Does Hebrews 10 speak of losing our salvation?

In his Hebrews study, Pastor Armstrong maintains that once a person is saved by faith in Jesus Christ he is forever saved. He cannot lose his salvation though he might lose his eternal rewards. However in Chapter 10 the writer to the Hebrews makes it clear that anyone who continually sins will lose his salvation. There is no further sacrifice that can save him. Which of the two positions is correct?

The passage you mention from Hebrews 10 is not speaking of a loss of salvation. The writer is reminding the reader of how seriously the Lord takes our relationship with Him:

Heb. 10:26 For if we go on sinning willfully after receiving the knowledge of the truth, there no longer remains a sacrifice for sins,
Heb. 10:27 but a terrifying expectation of judgment and THE FURY OF A FIRE WHICH WILL CONSUME THE ADVERSARIES.
Heb. 10:28 Anyone who has set aside the Law of Moses dies without mercy on the testimony of two or three witnesses.
Heb. 10:29 How much severer punishment do you think he will deserve who has trampled under foot the Son of God, and has regarded as unclean the blood of the covenant by which he was sanctified, and has insulted the Spirit of grace?
Heb. 10:30 For we know Him who said, “VENGEANCE IS MINE, I WILL REPAY.” And again, “THE LORD WILL JUDGE HIS PEOPLE.”
Heb. 10:31 It is a terrifying thing to fall into the hands of the living God.

The person in view here has been sanctified, so they are in the family of God by faith. Nevertheless, this person goes on sinning willfully, and therefore they place themselves in jeopardy of God's judgment. The judgment isn't a question of Heaven or hell but rather a judgment over rewards. The writer's reference to fire in v.27 is an allusion to the judgment seat of Christ, similar to Paul's words in 1 Corinthians 3:

1Cor. 3:14 If any man’s work which he has built on it remains, he will receive a reward.
1Cor. 3:15 If any man’s work is burned up, he will suffer loss; but he himself will be saved, yet so as through fire.

Believers should recognize that our judgment moment before God can be a terrifying thing if we arrive having lived a life of willful disobedience. It's precisely because we are in covenant with God that this moment is so serious. Notice the writer says the Lord will judge "His people" in v.30, which refers to the judgment seat of Christ for believers.

So this passage isn't suggesting that a believer could forfeit, lose or otherwise undo their salvation. It's a reminder that our obligations to serve God are serious, and the consequences for failing to do so are also serious.

In fact, to conclude that salvation can be undone is evidence that we don't understand how salvation takes place in the first place. For more information about how salvation happens and why it cannot be reversed, we strongly recommend you listen to our Romans Bible study, which is the Bible's comprehensive explanation on obtaining righteousness. You can find it here:

Romans (2017)