Scientific errors are important, because anyone today can verify that they are false with relatively little effort, especially when compared to Muhammad's audience in the 7th century.
To this end, Muhammad taught the falsehood that flies carry both diseases, and their corresponding cures, on either wing, so they should be fully submerged in a drink that they touch:
Sahih al-Bukhari 3320
Narrated Abu Huraira:
The Prophet (ﷺ) said "If a house fly falls in the drink of anyone of you, he should dip it (in the drink) and take it out, for one of its wings has a disease and the other has the cure for the disease."
This is also narrated in Sahih al-Bukhari 5782, Sunan Ibn Majah 3505, and Sunan Abi Dawud 3844. It is verifiably false, and an extremely dangerous teaching.
If the wings of a fly are analyzed, it will not be the case that one will have a disease, and the other will have the corresponding cure for that disease. One wing of a fly is not always sterile and disease-fighting, with the other being only diseased and devoid of cure. And, adding the entire body of the fly to a drink has just exposed a person to millions and millions of bacteria, making the situation worse than it was when only a wing was submerged. This is a dangerous false teaching from someone who demanded that others obey him as if he were God (Surah 4:80, Sahih al-Bukhari 2956-57), and never question any of his judgments (Surah 4:65, 33:36, 49:1).
Muhammad also taught the falsehood that Ajwa dates grant immunity against poison, and magic:
Sahih al-Bukhari 5445
Narrated Sa`d:
Allah's Messenger (ﷺ) said, "He who eats seven 'Ajwa dates every morning, will not be affected by poison or magic on the day he eats them."
This is also narrated in Sahih al-Bukhari 5768-69, 5779, Sahih Muslim 2047a-b, and Sunan Abi Dawud 3876.
Not only is this claim false, but it is also exceptionally ironic, because Muhammad was both bewitched and poisoned in his life, according to himself, and poisoning eventually contributed to his death (Sahih al-Bukhari 4428). So, either Muhammad did not personally follow this practice, or he did follow it, and it did not work.
Other of Muhammad's scientific falsehoods are covered in the following articles:
Many of these scientific falsehoods have the potential to cause harm, and yet Muhammad claimed divine revelation for them, and said these things while professing his judgments to be equivalent to God's (Surah 4:65, 4:80).
Many Muslims base their faith on the perfect preservation of the Quran. Even though preservation of a text does not prove that it is divine, there are many narrations from the early period of Islam testifying that the Quran has not been preserved.
The most concise example of this corruption would be the verses of breastfeeding an adult five times, which according to Aisha, were in the Quran, and recited at the time of the death of Muhammad, yet such verses are absent from any Quran today:
Sahih Muslim 1452a
'A'isha (Allah be pleased with her) reported that it had been revealed in the Holy Qur'an that ten clear sucklings make the marriage unlawful, then it was abrogated (and substituted) by five sucklings and Allah's Apostle (ﷺ) died, and it was before that time (found) in the Holy Qur'an (and recited by the Muslims).
The lost verses referenced above taught that a woman breastfeeding a man five times made him unlawful for marriage. Originally, it was revealed that ten sucklings made marriage unlawful, but for some reason, this provision in the Quran - which is supposed to be the eternal word of Allah - was modified, and it was reduced to five, and recited that way. Then, this ceased being recited completely, after the death of Muhammad.
The above is far from the only example of lost or altered portions of the Quran that are discussed in Muslim sources. Lost verses and chapters are mentioned repeatedly in various narrations:
The topic of the corruption of the Quran is covered in much more detail, in its own article here.
Muhammad taught a very imminent end of the world, and in turn, gave many prophecies of signs that were to accompany the Last Hour. He also gave narrations establishing a time boundary on when the Last Hour was to take place.
For instance, in the following narration, he set a time bound of 100 years for the Last Hour:
Sahih al-Bukhari 564
Narrated `Abdullah:
"One night Allah's Messenger (ﷺ) led us in the `Isha' prayer and that is the one called Al-`Atma by the people. After the completion of the prayer, he faced us and said, "Do you know the importance of this night? Nobody present on the surface of the earth tonight will be living after one hundred years from this night."
This is also narrated in Sahih al-Bukhari 116. The Arabic in the passage actually says "not a single person on the surface of the earth will remain", but that is obscured here by the Muslim translators in order to accommodate the Muslim defense of this failed prophecy, which is to say that it is Muhammad prophesying that the current earth's population would all be dead in 100 years.
This defense fails for a few reasons:
Any impartial observer can consider the above proposals, and determine that the Muslim defense of this passage is ad-hoc, and illogical. Muslims must understand that simply having "a response" means absolutely nothing, if that response is logically absurd, and uncompelling.
Elsewhere, Muhammad prophesied that the Last Hour would come within the lifetime of a young slave in his presence (Sahih al-Bukhari 6167), or within the lifespan of an Ansari child (Sahih Muslim 2952, 2953a-c). He frequently claimed that the Last Hour was very near (Sahih al-Bukhari 2222, 3448, 4936, 5301, 6503-05, Sahih Muslim 155a, 867a, 2950, 2951b-e). He even was open to a young man named Ibn Sayyad being the Dajjal, which is an End Times figure in Islam (Sahih al-Bukhari 3055, Sahih Muslim 169d, 2924a-b, 2925, 2930a-c, 2931), even spying on him (Sahih al-Bukhari 3056-57, Sahih Muslim 169d, 2930a-c, 2931), and causing much trouble for him (Sahih Muslim 2927a-c, 2929).
Perhaps the most absurd false prophecy that Muhammad gave pertaining to the Last Hour stems from the Quran itself (Surah 18:92-99, 21:95-97), where Muhammad tells of the vicious, savage tribes of Gog and Magog, which are currently kept at bay by Allah miraculously repairing an iron wall between two mountains every day, placed there originally by Zul-Qarnain. One day, they will be allowed to break through, and will attack all of the earth. This false prophecy is covered in its own article here.
Muhammad claimed that he was a continuation of the Biblical line of prophets (Surah 3:3-4), and claimed to be mentioned in the Torah and Gospel that people had at his time:
Surah 7:157
157 Those who follow the Messenger, the Unlettered Prophet, whom they find mentioned in the Torah and the Gospel in their possession. He directs them to righteousness, and deters them from evil, ...
It is known what the Torah and Gospel looked like during his time, because there are existing copies of the entire Bible from centuries before Muhammad, and also tens of thousands of quotations of the Bible in writers writing in many different languages, in many different countries, who likewise lived prior to Muhammad. Anyone with an internet connection can read these writers and verify this. They are called the Ante-Nicene, Nicene, and Post-Nicene writers, and we have thousands of pages of their writings.
However, in spite of Muhammad's claim, he is not mentioned anywhere in either the Torah, or the Gospel, let alone both of them. The Bible passages which Muslims use in an attempt to vindicate his claim in Surah 7:157 uniformly involve dishonest handling of the Scripture, and are addressed individually in the article Is Muhammad Prophesied in the Bible?. There is no way for a person who is sincere in their quest for the truth before God to remain a Muslim, if they undertake a serious examination of just this one claim of Muhammad's.
Not only is his claim to be mentioned in the Bible completely indefensible and false, but also the religion and God that he taught is entirely different from that of the Bible. He got even basic things, such as the name of God, completely wrong.
The name of God in Abrahamic tradition is Yahweh (Exodus 3:13-15), which is affirmed by the fact that it is embedded in many of the names of the prophets, such as Jesus ("Yahweh is salvation"), John ("Yahweh is gracious"), Elijah ("My God is Yahweh"), and so on - all of whom the Quran affirms as prophets. Yet, Muhammad never demonstrated an awareness of the significance of this name, instead favoring "Allah" as the proper name of God. Many such themes shared between the Old and New Testament, yet missing from Islam entirely, are covered in this article.
Similarly, despite his frequent claims of affirmation (Surah 2:40-41, 2:89, 5:43-47, 5:68, etc.), he contradicted what the Bible taught on many dozens of doctrines, affecting thousands of verses of Scripture. To give one example:
Surah 2:230
230 So if a husband divorces his wife, then it is not lawful for him to remarry her until after she has married another man and then is divorced. Then it is permissible for them to reunite, as long as they feel they are able to maintain the limits of Allah. These are the limits set by Allah, which He makes clear for people of knowledge.
Contrasted with:
Jeremiah 3:1
1 They say, If a man put away his wife, and she go from him, and become another man's, shall he return unto her again? shall not that land be greatly polluted? but thou hast played the harlot with many lovers; yet return again to me, saith the LORD.
Above, Muhammad taught that after a divorce, a woman must marry another man, before the original couple can be reconciled. The Bible says that such a practice causes the land to be "greatly polluted", and in the original giving of the Law, the Lord calls the practice an "abomination" (Deuteronomy 24:1-4).
Anyone can look at the above, and see that the Bible's ruling is more sensible than the Quran's, and there is no sensible motivation for anyone to "corrupt" Jeremiah 3:1 and Deuteronomy 24:1-4 to change from the Quran's nonsensical requirement of getting married to someone else, and divorcing them, before reconciliation is allowed. Rather, Muhammad simply made his own rule up, and the rule is ridiculous, and contradicts what God actually commanded.
Many other examples of Muhammad contradicting the Biblical prophets are covered in the following articles:
Additionally, while Muhammad did not affirm the Biblical prophets at all, he did incorporate the practices of Arabian Pagans into his religion. Some examples of this borrowing are covered in the article Examining the Pagan Origins of the Kaaba, the Black Stone, and Safa and Marwa, which cites narrations from Muhammad's companions stating that they had done the exact same practices before Islam when they lived as Pagans near Muhammad's hometown.
There is much more that could be said - see the linked articles - but in summary, Muhammad invalidated his own new religion, by claiming to affirm a prophetic tradition that he knew next to nothing about, and prolifically contradicted.
Muhammad falsely attributed revelations to God on many occasions - the aforementioned scientific falsehoods being one example.
However, his most egregious false revelation was when, rather early in his prophetic career, he recited revelations given to him by Satan, according to himself. During this Satanic Verses incident, Muhammad is said to have given the most damning confession that any self-proclaimed prophet could give:
The History of al-Tabari, Vol 6, pg. 111
That evening Gabriel came to him and reviewed the surah with him, and when he reached the two phrases which Satan had cast upon his tongue he said, "I did not bring you these two." Then the Messenger of God said, "I have fabricated things against God, and have imputed to Him words which He has not spoken."
In the Bible, the Lord was clear in stating that this type of thing was never to happen to a true prophet. Anyone who did what Muhammad did, according to his own testimony, would immediately be put to death under the Law of God:
Deuteronomy 18:20
20 But the prophet, which shall presume to speak a word in my name, which I have not commanded him to speak, or that shall speak in the name of other gods, even that prophet shall die.
Apart from the Satanic Verses incident, and all of the prolific contradictions of science, medicine, and previous Scripture that he claimed to affirm, he also gave "revelations" which were all-too-human in nature and purpose. Consider the following:
In addition, the Quran contains stories which are known to originate from non-inspired Jewish commentary on the Bible, Christian apocrypha, as well as a legendary biography of Alexander the Great. Some articles covering this are as follows:
Finally, he put words into people's mouths prolifically, and made everyone in his book sound exactly like him. This phenomenon is covered in its own article here.
Muhammad's revelations were not divine. They were very human, very fallible, and often, very ridiculous, or altogether false.
Taken on their own, any one of these points invalidate Muhammad as a prophet. Taken together, they leave no excuse for an intellectually honest Muslim to stay in Islam.
Muhammad was just one of many false prophets that Jesus Christ warned about (Matthew 24:11). Unlike Muhammad, Jesus could actually affirm the revelation of all the prophets that came before Him (Matthew 5:17), because they all prophesied of Him (Acts 3:18).