Matthew 3:10
10 And now also the axe is laid unto the root of the trees: therefore every tree which bringeth not forth good fruit is hewn down, and cast into the fire.
Luke 3:9
9 And now also the axe is laid unto the root of the trees: every tree therefore which bringeth not forth good fruit is hewn down, and cast into the fire.
These verses are used occasionally as proof texts for the idea that if one doesn't do enough good works, they will go to Hell.
The assertion is:
Let's examine whether that's a valid interpretation of these passages.
These parallel passages are quotations from John the Baptist, preaching to the Pharisees and Sadducees, as the context demonstrates:
Matthew 3:7-10
7 But when he saw many of the Pharisees and Sadducees come to his baptism, he said unto them, O generation of vipers, who hath warned you to flee from the wrath to come?
8 Bring forth therefore fruits meet for repentance:
9 And think not to say within yourselves, We have Abraham to our father: for I say unto you, that God is able of these stones to raise up children unto Abraham.
10 And now also the axe is laid unto the root of the trees: therefore every tree which bringeth not forth good fruit is hewn down, and cast into the fire.
Luke 3:7-9
7 Then said he to the multitude that came forth to be baptized of him, O generation of vipers, who hath warned you to flee from the wrath to come?
8 Bring forth therefore fruits worthy of repentance, and begin not to say within yourselves, We have Abraham to our father: for I say unto you, That God is able of these stones to raise up children unto Abraham.
9 And now also the axe is laid unto the root of the trees: every tree therefore which bringeth not forth good fruit is hewn down, and cast into the fire.
In the passages, John mentions mentions that their trust in their physical Jewish lineage - "we have Abraham to our father" - which they thought was sufficient to justify them. It's only after mentioning this fact that we get the mention of the "axe being laid" to the trees.
And notice, John says "now" the axe is laid to the trees. What was happening "now" when John was preaching? It was the beginning of Christ's ministry - which is why both passages go on to mention how "one mightier than I cometh", who will be the one to determine which "trees" are spared the "fire", or who is "chaff" as opposed to "wheat" (Matthew 3:11-12, Luke 3:16-17).
Therefore, one primary application of the passages - probably the most obvious application - is to tell the Pharisees that unless they individually believe ("repent") in the Messiah who is coming "now", they will go to Hell, and unless they "bring forth fruits worthy of repentance", they will suffer physical destruction as well, as the Christ who is coming will not tolerate rebellion in His kingdom. Their heritage alone will not avail them, because God is "not a respecter of persons" (Acts 10:34, Romans 2:11).
Along these lines, it's important to point out that this preaching is in reference to what was, at the time, the preaching of an imminent kingdom. If the Jews as a people had accepted Jesus as their Messiah, He would have begun the Millennium right then and there, which is why both He and John preached about the Kingdom of Heaven being "at hand" (Matthew 3:2, 4:17, 10:7, Mark 1:15).
However, because the Jews crucified Him instead (1 Thessalonians 2:14-15, Acts 4:8-10, etc.), we get the other primary application of the passage - one of the major themes in the New Testament - which is the Jews as God's chosen people being replaced ("hewn down") with a mixed multitude of Jew and Gentile believers in the Messiah, and having their physical nation taken from them - which overtly came to pass in the First Jewish-Roman War, around 70 A.D., when Jerusalem was sacked ("cast into the fire"). This relates to the Temple in Jerusalem being destroyed - another symbol of God replacing the physical Jews - which Christ prophesied multiple times (Matthew 24:1-2, Mark 13:1-2, Luke 19:41-44, 21:5-6).
Elsewhere in Scripture, we see that the Kingdom that the Jews would have inherited had they accepted Jesus as the Messiah as a group, has been taken from them, and given to the spiritual nation of believers:
Matthew 21:43
43 Therefore say I unto you, The kingdom of God shall be taken from you, and given to a nation bringing forth the fruits thereof.
This theme of the Jews being replaced by a spiritual nation consisting of all believers as the chosen people of God is covered in more detail in an article titled Christians are God's Chosen People, Not Jews. It is something many of the Lord's parables dealt with (Matthew 21:28-41, 22:1-14 Mark 12:1-9, Luke 16:19-31, 20:9-16, etc.).
In summary, whether individually, or as a nation, the only way for the "trees" to "bring forth fruit" is to believe in Jesus Christ, and "not to say within (themselves), We have Abraham to our father". That is the primary application of the passages, and they are directed towards unbelieving Jews who were not open to the imminent coming of the Messiah, Jesus.
Related to the primary message of the passages in question is a general principle that applies even to believers, who have Eternal Life.
The principle is that, as Jesus says in John chapter 15, that if a believer doesn't abide in Christ, they too will be broken off from the "vine", which is Jesus Christ, "wither", and are "burned":
John 15:1-8
1 I am the true vine, and my Father is the husbandman.
2 Every branch in me that beareth not fruit he taketh away: and every branch that beareth fruit, he purgeth it, that it may bring forth more fruit.
3 Now ye are clean through the word which I have spoken unto you.
4 Abide in me, and I in you. As the branch cannot bear fruit of itself, except it abide in the vine; no more can ye, except ye abide in me.
5 I am the vine, ye are the branches: He that abideth in me, and I in him, the same bringeth forth much fruit: for without me ye can do nothing.
6 If a man abide not in me, he is cast forth as a branch, and is withered; and men gather them, and cast them into the fire, and they are burned.
7 If ye abide in me, and my words abide in you, ye shall ask what ye will, and it shall be done unto you.
8 Herein is my Father glorified, that ye bear much fruit; so shall ye be my disciples.
What happens to branches that break off from the main trunk of a tree? They dry out, and then men gather them up, and burn them.
For a Christian, this "withering" represents walking in the flesh, and this "burning" represents the physical destruction experienced from the chastening of the Lord, because "whom the Lord loveth he chasteneth, and scourgeth every son whom he receiveth" (Hebrews 12:6):
1 Corinthians 5:5
5 To deliver such an one unto Satan for the destruction of the flesh, that the spirit may be saved in the day of the Lord Jesus.
The only destruction a Christian ever need fear is physical, because we already have assurance that we "shall never die" (John 11:26) and "shall never perish" (John 10:28) in the spiritual sense, as is discussed in many other articles on this site. However, if Christians "live after the flesh", they still are said to "die" (Romans 8:12-13), because they are walking their old nature - "the body of this death" (Romans 7:24). They spiritually, and often physically, "wither", and we have an example of God eventually killing disobedient believers (1 Corinthians 11:30-32).
Reading "Hell" into the "burning" mentioned in John 15 would require simply ignoring hundreds of passages of Scripture, which teach that once someone believes in Christ, they have Everlasting Life (John 6:47, etc.), and a host of other things (see this article). God's judgment, even on Christians, is symbolized by fire (1 Corinthians 3:11-15, Hebrews 6:7-8, 10:26-27).
In conclusion, these parallel passages in Matthew and Luke teach a truth both for the lost, and the saved - the only way to avoid spiritual and physical destruction here in this life is to believe in, and afterwards abide in, Jesus Christ.
The penalty for not believing, though, is different from the one for not abiding. For not believing, one goes to Hell (John 3:18, 3:36, etc.), and often also faces temporal judgment - exemplified by the Jews having their nation taken from them and their capital destroyed as a result of not believing in Jesus as the Messiah. The penalty for a believer not abiding, on the other hand, is only ever limited to physical chastening, and loss of rewards at the Judgment Seat of Christ.