Multiple times in Christ's ministry, He taught on the need to forsake all in order to become His disciple. This is found in:
Clearly, this is something that the Lord wanted to emphasize, and since these passages are sometimes used to teach a works-based Salvation, it's vital to have a good understanding of what the Lord is teaching in them.
Because these verses cover the same topic, and use very similar terms, we can look at them all together, and then summarize their teaching:
Matthew 10:37-39
37 He that loveth father or mother more than me is not worthy of me: and he that loveth son or daughter more than me is not worthy of me.
38 And he that taketh not his cross, and followeth after me, is not worthy of me.
39 He that findeth his life shall lose it: and he that loseth his life for my sake shall find it.
Matthew 16:24-26
24 Then said Jesus unto his disciples, If any man will come after me, let him deny himself, and take up his cross, and follow me.
25 For whosoever will save his life shall lose it: and whosoever will lose his life for my sake shall find it.
26 For what is a man profited, if he shall gain the whole world, and lose his own soul? or what shall a man give in exchange for his soul?
Mark 8:34-38
34 And when he had called the people unto him with his disciples also, he said unto them, Whosoever will come after me, let him deny himself, and take up his cross, and follow me.
35 For whosoever will save his life shall lose it; but whosoever shall lose his life for my sake and the gospel's, the same shall save it.
36 For what shall it profit a man, if he shall gain the whole world, and lose his own soul?
37 Or what shall a man give in exchange for his soul?
38 Whosoever therefore shall be ashamed of me and of my words in this adulterous and sinful generation; of him also shall the Son of man be ashamed, when he cometh in the glory of his Father with the holy angels.
Luke 9:23-26
23 And he said to them all, If any man will come after me, let him deny himself, and take up his cross daily, and follow me.
24 For whosoever will save his life shall lose it: but whosoever will lose his life for my sake, the same shall save it.
25 For what is a man advantaged, if he gain the whole world, and lose himself, or be cast away?
26 For whosoever shall be ashamed of me and of my words, of him shall the Son of man be ashamed, when he shall come in his own glory, and in his Father's, and of the holy angels.
Luke 14:26-27, 33
26 If any man come to me, and hate not his father, and mother, and wife, and children, and brethren, and sisters, yea, and his own life also, he cannot be my disciple.
27 And whosoever doth not bear his cross, and come after me, cannot be my disciple.
33 So likewise, whosoever he be of you that forsaketh not all that he hath, he cannot be my disciple.
Luke 17:33
33 Whosoever shall seek to save his life shall lose it; and whosoever shall lose his life shall preserve it.
John 12:25-26
25 He that loveth his life shall lose it; and he that hateth his life in this world shall keep it unto life eternal.
26 If any man serve me, let him follow me; and where I am, there shall also my servant be: if any man serve me, him will my Father honour.
From the above, we see that:
Clearly, the cost of being a disciple of Christ requires the ultimate and total sacrifice of one's own life. That is not where the contention lies. The contention is found in the fact that nearly every branch of professing Christianity sees this as a requirement for Salvation.
Such take these verses, and then assert that if you aren't a disciple, you aren't saved. This is put concisely in what has become a cliché and the calling card of "Lordship Salvation" - "if He's not Lord of all, He isn't Lord at all" - i.e. - unless you forsake all to follow Jesus, you will go to Hell when you die.
This argument is based on the consequences Christ gives for not being a disciple, and so understanding those is the linchpin to having a proper view of these passages.
Firstly, what are the "positive" consequences for those who forsake all to become disciples? They are said to "save" (Mark 8:35, Luke 9:24), "preserve" (Luke 17:33), or "find" (Matthew 10:39, 16:25) their life, "keeping it unto life eternal" (John 12:25).
This means that their lives will have eternal significance, rather than being whittled away in vanity. The only way to "save" one's life, according to the passages, is to not "save" it. This is in reference to "saving it for yourself" by trying to hang onto the world, and the carnal desires of the flesh, and making that your life's goal, as we are warned against elsewhere in Scripture (1 John 2:15, etc.).
Discipleship has eternal ramifications because of those we may win to Christ, and also because disciples stand to earn eternal rewards, which will abound long after their physical life is over, something covered in depth in another article. Being a disciple of Jesus Christ is the only way for someone to avoid living a pointless life, and instead "find" a life of meaning.
On the other hand, the "negative" consequences for those who don't become disciples are "losing their life" (Matthew 10:39, 16:25, Mark 8:35, Luke 9:24, John 12:25), and, as is implied, such "lose their soul" (Matthew 16:26, Mark 8:36-37), defined in the parallel passage as "losing himself" (Luke 9:25), or being "cast away" (Luke 9:25).
This teaches the same thing as the positive consequences - anyone who lives for themselves here in this life, is living pointlessly. They are losing any chance to live a life of eternal significance.
The mention of losing one's "soul" is what is typically cited as proof that non-disciples will go to Hell, but as mentioned twice already, this is defined as "losing oneself" in the parallel passage in Luke (Luke 9:25), and the word "soul" often just means "life" (Genesis 12:5, 12:13, Joshua 10:35, 39, Job 12:10, Jeremiah 2:34, Lamentations 1:19, Ezekiel 7:19, 13:18-20, Luke 12:20, 21:19, Acts 3:23, 7:14, 27:37, Revelation 16:3, 18:13, etc.), with the same Greek word being translated "life" in the verses immediately before both of the references to losing one's "soul" (Matthew 16:25, Mark 8:35). This is teaching the same thing as the other passages - by not being a disciple, you lose the life and identity that you could have had if you had chosen to follow Christ.
The fact that the standard for discipleship is so high ensures that there is not a single human being on this planet that is a disciple of Christ at all times. Whenever you sin, you are not being a disciple, because discipleship involves forsaking "all" (Luke 14:33). Complete discipleship, where Jesus is Lord of all aspects of your life, would require sinless perfectionism, unless "all" has been redefined to mean "some, sometimes".
Discipleship, then, unlike Salvation, is not binary. It's on a spectrum, and waxes or wanes in accordance with how well a believer is walking in the New Man. Even believers whose lives can be generally characterized by being a disciple fall short of total discipleship every hour of every day, meaning that if it were a requirement for Salvation, everyone would go to Hell.
The only way to make this a requirement for Salvation without condemning everyone is to lower Christ's actual standard of discipleship from sinless perfection (Matthew 5:48) and total surrender of "all" (Luke 14:33) to some nebulous level of partial commitment and obedience. That isn't what Christ demands in the passage, but rather an attempt by a self-righteous sinner to bring Christ down to their level, so that they can feel like they've earned their way to Heaven.
And of course, it should go without mentioning that if Salvation were dependent on being a disciple, given what being a disciple entails, it would be a laughable lie to ever say that Salvation is a "free gift" (Romans 5:15-18), and "not of works" (Ephesians 2:8-9). It doesn't take "work" to "forsake all"? Really? On what planet? And why mention it, then, if it's automatic?
Discipleship is preached on repeatedly precisely because it's not automatic, nor a guarantee, and requires extreme amounts of effort on behalf of the believer, "daily" (Luke 9:23), which is why it has nothing to do with whether they've been saved from Hell by faith in Jesus Christ.
Salvation is a free gift obtained by simple faith in Jesus Christ as Savior (Acts 16:30-31), as is covered in the article Eternal Life is Called a Gift for a Reason. Discipleship, on the other hand, is a lifelong struggle to follow Christ in deed, after Salvation.
Conflating the two terms not only teaches an extreme, cut-and-dry works Salvation, but also condemns every single person on the planet, as none consistently live up to the standard that Christ expects of us.