There are two questions here:
- Meaning of ζημιόω (zémioó)
According to BDAG, ζημιόω (zémioó) means:
- to experience the loss of something, with implication of undergoing hardship or suffering, suffer damage/loss, forfeit, sustain injury, eg, Matt 16:26, Mark 8:36, Luke 9:25, 2 Cor 7:9, Phil 3:8
- to be punished, eg, 1 Cor 3:15
Some interpreters would say 1 Cor 3:15 belongs in the first category and thus argue for a single meaning in all instances.
- Meaning in 1 Cor 3:15
Paul's intended meaning in 1 Cor 3:15 cannot be divorced from the previous verses of which V15 is merely a conclusion:
12 If anyone builds on this foundation using gold, silver, precious stones, wood, hay, or straw, 13 his workmanship will be
evident, because the Day will bring it to light. It will be revealed
with fire, and the fire will prove the quality of each man’s work.
14 If what he has built survives, he will receive a reward. 15 If it is burned up, he will suffer loss. He himself will be saved, but
only as if through the flames.
Note that Paul is using the metaphor for salvation of the foundry refining metal such as gold. The standard practice for refining gold is to heat a mixture - the pure metal would be molten and drain away and the unwanted dross burned up. Thus, the pure metal would escape through the fire.
Thus, in the Christian life of the Christian teacher (see V9-11), those who have built/taught well with precious stone using "gold" will have their work survive; while those who have used cheap material such as wood and hay will have their work burned up and suffer loss.
Note the comments of Ellicott in V 14, 15 -
(14) This is the general application to Christian teachers of what has
gone before. Those who have built well shall have their reward in
their work having survived the trial of the fire; those who have built
otherwise shall lose everything—their work, which should have remained
as their reward, will perish in the fire—and they themselves will be
as men who only make good their escape by rushing through a
conflagration, leaving all that was theirs to be destroyed. (See Mark
9:49.)
(15) So as.—These words remind us that the whole passage, and
especially the reference to fire, is to be regarded as metaphorical,
and not to be understood in a literal and physical sense. Forgetting
this, Roman divines have evolved from these words the doctrine of
purgatory.
Thus, the "fire" in Paul's metaphor is what happens in the light of the judgement - when all is revealed, some people's teaching/building will have been shown to be worthless and destroyed and others will have been shown to be valuable. That is, it not the reward of the righteous in view but the value of the work each Christian teacher does. The reward is to see what contribution the person has made.
The relevance of Matt 20:10 is that all these people get to heaven (the denarius) but their rewards will be different because the judgement will reveal whether they have been teaching truth or error. Benson sums this well:
1 Corinthians 3:14-15. If any maws work abide which he hath built, &c. — If the superstructure which any minister of Christ raises on the
true foundation, if the doctrines which he preaches can bear the test
by which they shall be tried at that day, as being true, important,
and adapted to the state of his hearers; and the converts which he
makes by preaching these doctrines, be of the right kind, truly
regenerated and holy persons, he shall receive a reward — In
proportion to his labours. If any man’s work shall be burned — If the
doctrines which any minister preaches cannot bear the test of the
great day, as being false or trivial, or not calculated to convert and
edify his hearers; or if the converts which he makes by preaching such
doctrines be only converts to some particular opinion, or mode of
worship, or form of church government, or to a certain sect or party,
and not converts to Christ and true Christianity, to the power as well
as the form of godliness, to the experience and practice, as well as
to the theory of true religion, and therefore cannot stand in that
awful judgment, he shall suffer loss — Shall lose his labour and
expectation, and the future reward he might have received, if he had
built with proper materials; as a man suffers loss who bestows his
time and labour on the erection of a fabric of wood, hay, and stubble,
which is afterward consumed. But he himself — That preacher himself;
shall be saved — Supposing he himself be a true disciple of Christ,
built up in faith and holiness on the true foundation; yet so as by
fire — As narrowly as a man escapes through the fire, when his house
is all in flames about him: or rather, if so be that his own religion,
his personal faith and holiness, can bear both the fiery trial which
he may be called to pass through on earth, whether of reproach and
persecution, or of pain and affliction, or any other trouble, and also
the decisive trial of the last day.
Recall that Jesus says this in Rev 22:12 -
“Behold, I am coming soon, and My reward is with Me, to give to each
one according to what he has done.
See also Gal 6:7, 8, Rom 2:6, Matt 16:27, Isa 62:11, 40:10, etc.