For if someone with a weak conscience sees you, with all your knowledge, eating in an idol's temple, won't that person be emboldened to eat what is sacrificed to idols? So this weak brother or sister, for whom Christ died, is destroyed by your knowledge. (1 Cor 8:10-11 NIV)
1 Cor 8 is a chapter about the question of whether it is permissible for a Christian to eat food that had been offered to pagan deities (v.1). Paul allows for strong-willed believers to consume such meat, as such idols have no ontological existence (v.4). However, for a “weak” Christian, one is to refrain from eating such in their presence, as it could upset or even offend their sensibilities, and perhaps even lead to apostasy. The language used of such people in vv.10-11 is not that of a superficial believer who only had a false confession of faith and was never truly converted; indeed, the phrase in v.11 is ὁ ἀδελφὸς δι᾽ ὃν Χριστὸς ἀπέθανεν. The preposition δια is used in conjunction with ον, the pronoun meaning "who" and is in the masculine nominative, relating back to the phrase, "the brother" (ο αδελφος). Δια coupled with the accusative means "on behalf of," so this verse means exactly what our English translations mean--Christ died for/on account of this person who is said to be "destroyed."
A parallel text is the following:
But if thy brother be grieved with thy meat, now walkest thou not charitably. Destroy not him with thy meat, for whom Christ died. (Rom 14:15)
In the underlying Greek, Paul states that Christ died “for” (υπερ) the Christian who could be “destroyed.” This preposition is a favourite of Paul’s to denote those for whom Christ died (Rom 5:6, 8; 8:34; 1 Cor 15:3; 2 Coe 5:20; Gal 2:20; Eph 5:2, 25). This itself calls into serious question the doctrine of “particular redemption” or “Limited Atonement.”
The verb "to destroy" is απολλυμι, and means to die/destroy/perish, and if often used in eschatological contexts, referring to eternal damnation. Note the usages of the verb in the Pauline epistles:
For as many have sinned without law shall also perish (απολλυμι) without law: and as many as have sinned in the law shall be judged by the law. (Rom 2:12)
For the preaching of the cross is to them that perish (απολλυμι) foolishness; but unto us which are saved it is the power of God. For it is written, I will destroy (απολλυμι) the wisdom of the wise, and will bring to nothing the understanding of the prudent. (1 Cor 1:18-19)
Neither let us tempt Christ, as some of them also tempted, and were destroyed (απολλυμι) of serpents. Neither murmur ye, as some of them also murmured, and were destroyed (απολλυμι) of the destroyer. (1 Cor 10:9-10)
[If the resurrection of Jesus never happened] They they also which are fallen asleep in Christ are perished (απολλυμι). (1 Cor 15:18)
For we are unto God as a sweet savour of Christ, in them that are saved, and in them that perish (απολλυμι): To the one we are the savour of death unto death; and unto the other the savour of life unto life. And who is sufficient for these things? (2 Cor 2:15-16)
But if our gospel be hid, it is hid to them that are lost (απολλυμι) . . . Persecuted, but not forsaken; cast down, but not destroyed (απολλυμι) (2 Cor 4:3, 9)
And with all deceivableness of unrighteousness in them that perish (απολλυμι); because they received not the love of truth, that they might be saved. (2 Thess 2:10)
In light of the terminology Paul uses in 1 Cor 8:10-11 (cf. Rom 1415):, it is perfectly understandable to see why this is seen, correctly, as another piece of biblical evidence refuting (1) limited atonement and (2) various theologies of eternal security, as this person is (i) one for whom Christ died for; (ii) presented as being a true Christian, and (iii) faces eschatological destruction/damnation due to their apostasy, not merely ecclesiastical disciple or temporary loss of fellowship.