What does the phrase "saved as by fire" in 1 Corinthians 3:15 mean?
If any man's work shall be burned, he shall suffer loss: but he himself shall be saved; yet so as by fire.
An Evangelical perspective:
This verse should be understood in its context (the whole chapter) and a key verse for putting it in the right perspective is verse 13:
their work will be clearly seen, because the Day of Judgment will make it visible. That Day will appear with fire, and the fire will test everyone’s work to show what sort of work it was. - 1 Corinthians 3:13 NCV (emphasis added)
This passage of scripture refers to what is known as the Bema seat judgment. The Bema seat judgment is where believers will give an account for their works and be judged and rewarded accordingly, which is distinct to the type of judgment rendered at the Great White Throne judgment which separates believers unto salvation from unbelievers unto destruction.
If a believer's works are completely judged as worthless even though they have trusted in Christ for their own salvation, it will be like they have been snatched from a fire - they themselves are safe, but everything they labored for is destroyed and unsalvageable. There will of course be relief at being saved, but also a sense of loss as the things labored for that were held dear by the one being judged are revealed as insubstantial, temporary and not 'built' according to the will of the Master.
Most commentaries point toward the idea that the individual loses all, and because of this, is convinced of the need of salvation. Put another way, trials and tribulations make the individual realize the need for Christ. They burn away all the non-essentials, and leave only that which is pure and true.
Some examples to back up that answer:
Gill's Exposition of the Entire Bible
yet so as by fire; with much difficulty, and will be scarcely saved; see 1 Peter 4:17 with great danger, loss, and shame; as a man that is burnt out of house and home, he escapes himself with his own life, but loses all about him: so the Syriac version reads it, , "as out of the fire": see Zechariah 3:2. Or the sense is, that he shall be tried by the fire of the word, and convinced by the light of it of the errors, irregularities, and inconsistencies of his ministry;
Barnes' Notes on the Bible
Yet so as by fire - ὡς διὰ πυρός hōs dia puros. This passage has greatly perplexed commentators; but probably without any good reason. The apostle does not say that Christians will be doomed to the fires of purgatory; nor that they will pass through fire; nor that they will be exposed to pains and punishment at all; but he "simply carries out the figure" which he commenced, and says that they will be saved, as if the action of fire had been felt on the edifice on which he is speaking. That is, as fire would consume the wood, hay, and stubble, so on the great Day everything that is erroneous and imperfect in Christiana shall be removed, and that which is true and genuine shall be preserved as if it had passed through fire. Their whole character and opinions shall be investigated; and that which is good shall be approved; and that which is false and erroneous be removed.
Matthew Poole's Commentary
But if his work do not abide, if it shall appear upon the more clear and bright shining out of the truth of the gospel, that though he hath held the foundation right, yet he hath built upon it wood, hay, and stubble, mixed fables, and idle stories, and corrupt doctrine with the doctrine of the gospel,
he shall suffer loss by it, either by the afflicting hand of God, or by a loss of his reputation, or some other way. But yet God will not cast off a soul for every such error, if he keeps to the main foundation, Jesus Christ; he shall be saved, though it be as by fire, that is, with difficulty; which certainly is a more natural sense of this text, than those give, who interpret as by fire, of the fire of the gospel, or the fire of purgatory, of which the papists understand it
Several more commentaries can be found here.
A Catholic Sacred Scripture Interpretation
bruisedreed's answer has the gist of it.
1 Cor 3:11-15 (RSVCE) 11 For no other foundation can any one lay than that which is laid, which is Jesus Christ. 12 Now if any one builds on the foundation with gold, silver, precious stones, wood, hay, stubble— 13 each man’s work will become manifest; for the Day[a] will disclose it, because it will be revealed with fire, and the fire will test what sort of work each one has done. 14 If the work which any man has built on the foundation survives, he will receive a reward. 15 If any man’s work is burned up, he will suffer loss, though he himself will be saved, but only as through fire.
a. 3.13 the Day: i.e., the day of the Lord: God’s searching judgment.
The Navarre Bible New Testament Compact Edition has, in part, the following note on vv. 4-23:
The root of all apostolic work is God who gives growth (v.7). Man is God's instrument - servant, minister (v.5), or fellow-worker (v.9) - in that task, which can only be carried through if Jesus Christ is its foundation (v.11). Paul develops these ideas with the help of two similes - God's field (vv.6-9) and his building (vv.9-17). If Christ is the foundation, then Christians must not only be linked to Jesus Christ: they must, as it were, be glued to him [...]. He is the foundation and we are the building; he is the trunk of the vine and we are the branches; he is the husband and we are the wife; he is the shepherd and we are the flock (St. John Chrysostom, Hom. on 1 Cor 8:4). Extending the building metaphor, Paul appeals to ministers to act responsibly, reminding them that a Day of Judgment will come, the day of the LORD (vv.10-17).
Catholic Holy Tradition Perspective
From scripture, 'fire' is the means by which God tests and purifies his elect [cf.Hello's answer] and also the means he executes his searching judgment (see above). From Catholic Tradition, in this passage, the person has passed on, and that person is gold but impure, therefore they undergo purification by fire before admittance to the Beatific Vision. It is a cleansing fire.
We know the worst for a person and their achievements is when they gain the whole world and forfeit their life. The works in the OP's question that get burned up, are those that did not profit one as regards eternal life.
cf. This answer to According to Catholicism, in what cases is Purgatory necessary?
CCC 1031 The Church gives the name Purgatory to this final purification of the elect, which is entirely different from the punishment of the damned.1 The Church formulated her doctrine of faith on Purgatory especially at the Councils of Florence and Trent. The tradition of the Church, by reference to certain texts of Scripture, speaks of a cleansing fire:2
As for certain lesser faults, we must believe that, before the Final Judgment, there is a purifying fire. He who is truth says that whoever utters blasphemy against the Holy Spirit will be pardoned neither in this age nor in the age to come. From this sentence we understand that certain offenses can be forgiven in this age, but certain others in the age to come.3
1. cf. Council of Florence (1439):DS 1304; Council of Trent (1563):DS 1820; (1547):1580; see also Benedict XII, Benedictus Deus (1336):DS 1000.
2. cf. 1 Cor 3:15; 1 Pet 1:7.
3. St. Gregory the Great, Dial. 4,39:PL 77,396; cf. Mt 12:31.
cf. Wis 3:5-7 (RSVCE) 5 Having been disciplined a little, they will receive great good, because God tested them and found them worthy of himself; 6 like gold in the furnace he tried them, and like a sacrificial burnt offering he accepted them. 7 In the time of their visitation they will shine forth, and will run like sparks through the stubble.
RadzMatthewCoBrown asks: "What does the phrase "saved as by fire" in 1 Corinthians 3:15 mean? If any man's work shall be burned, he shall suffer loss: but he himself shall be saved; yet so as by fire."
1 Cor 3:15 is a standard verse used by Catholic apologists to support their idea of purgatory. They say this verse spells out that the 'man' in question's works were not of sufficient quality to withstand the test, so he had nothing (Catholic term, he had no merits) left after his works were tested, and he himself was saved very narrowly, only by "fire" - the purging fire of purgatory.
For and against purgatory has been argued by some Protestants(against) and some Catholics and others(for) - e.g. Luther, who continued to believed in purgatory even after he left the Catholic Church.
This scripture in itself can't prove purgatory exists.
One interpretation from Christians who don't believe in Purgatory is that this man's works were not of a quality that could stand the purity test from God, so a fire was sent to devour his 'works', and consumed his works so well, that the man just narrowly escaped the fire himself.
Old Testament has an example of purifying fire of the Lord in it, and the picture is of the Lord sitting as a goldsmith, and doing a work of refining the gold to remove impurities.
He will sit as a refiner and purifier of silver; he will purify the Levites and refine them like gold and silver. Then the LORD will have men who will bring offerings in righteousness[.]
The Levites and we Christians1 today, are both priests.