2 Corinthians 7:10
10 For godly sorrow worketh repentance to salvation not to be repented of: but the sorrow of the world worketh death.
This passage is used to teach that sorrow, and turning from sins, are necessary for Salvation from Hell. Does this verse teach that?
In the passage in question, "repentance" refers to turning from sin, as we can see that the context is the Corinthians becoming sorrowful, and repenting, upon receiving an earlier letter from the Apostle Paul:
2 Corinthians 7:8-11
8 For though I made you sorry with a letter, I do not repent, though I did repent: for I perceive that the same epistle hath made you sorry, though it were but for a season.
9 Now I rejoice, not that ye were made sorry, but that ye sorrowed to repentance: for ye were made sorry after a godly manner, that ye might receive damage by us in nothing.
10 For godly sorrow worketh repentance to salvation not to be repented of: but the sorrow of the world worketh death.
11 For behold this selfsame thing, that ye sorrowed after a godly sort, what carefulness it wrought in you, yea, what clearing of yourselves, yea, what indignation, yea, what fear, yea, what vehement desire, yea, what zeal, yea, what revenge! In all things ye have approved yourselves to be clear in this matter.
The above, especially verse 11, describes the repenting of sins by people who already believe in Jesus.
So, we can agree with the detractors of Free Grace theology on the first point - this verse definitely describes sorrow over, and turning from, sin.
However, to the point of contention - is the "salvation" being spoken of referring to Regeneration, or deliverance from temporal judgment? The same Greek word, "soteria", is used in both senses. For instance, the following uses of the same Greek word refer to what is clearly temporal deliverance:
Romans 13:11
11 And that, knowing the time, that now it is high time to awake out of sleep: for now is our salvation nearer than when we believed.
Philippians 1:19
19 For I know that this shall turn to my salvation through your prayer, and the supply of the Spirit of Jesus Christ,
Luke 1:71
71 That we should be saved from our enemies, and from the hand of all that hate us;
As the above demonstrate, context needs to be used to determine in what sense the word soteria is being used.
Since in our verse in question, the context is people who already believe turning from sin in response to a letter sent to their church, we can conclude that this is salvation from the temporal judgment of God on wayward believers (Hebrews 12:6), which is spoken of explicitly in another letter to the Corinthian church (1 Corinthians 11:29-32).
They already have eternal Salvation from their sins, since a person gets Eternal Life as a gift (Romans 6:23) the very instant they believe in Jesus as their Savior (John 6:47), before doing the difficult work of repenting of their sins and maintaining that state (Ephesians 2:8-10).
If, on the other hand, the meaning of this verse is taken from being a reference to things such as deliverance or blessings, which can be on a gradient, to something binary like whether one goes to Heaven or Hell, we would need to define just how sorrowful one must be, and just how many sins need to be repented of, and how diligently that state must be maintained, in order to qualify for Heaven. Since defining such things is impossible without making something up - no such cut offs below perfection are defined anywhere in Scripture - interpreting this verse in such a way makes assurance of Salvation impossible, and demolishes any hope or good news to be found in the Gospel.
In summary, the faulty application of this verse to refer to doing works, and feeling sorrow, in order to get to Heaven, comes from a misunderstanding of how the Bible uses the Greek word soteria. Similar mistakes are made with the word "sozo", also often translated "save" or "saved". They do not always refer to eternal Salvation from sins (Matthew 1:21), but often to deliverance from temporal peril or judgement.