Universalism, the broad movement and doctrinal system holding out the hope that every single person without exception will in the end repent and so be saved from Hell (whether in this life or after second death), is a faith relying heavily upon the notion that Christ’s substitution for sin on the cross was the payment of just punishment for the sins of every single person, also without exception. Consequently, if Christ died for absolutely everybody, it follows that Christ will also effect the salvation of absolutely everybody.
If Christ was punished for the sins of those who suffer in Hell, then their Hell need not necessary serve to satisfy God’s justice, for that has already been done. And it would be unnecessary for God to satisfy his justice a second time. In fact, it would be a contradiction, for God’s justice would not be propitiated (turned aside) at all if those for whom God ceases to be angry were then were to pour his wrath against their sin a second time, this time not on his Son, but now on those for whom his Son had already suffered ‘once for all’.
Universalists are forced to innovate to the vague notion that those under judgment in Hell, rather than paying themselves the sentence of God’s justice against their sins then and there, instead, contending that God meters out punishment in Hell primarily as a process of discipline that would turn eventually even the hardest of hearts to have faith in Christ, and so in the end empty Hell of every soul.
[1 Timothy 4:10]
1 Timothy 4:10 is an example a text used to rally support: “We have our hope set on the living God, who is the Savior of all people, especially of those who believe.” This verse is used to claim that if God is the Savior of every single person, he will in the end save every single person from Hell, whether before or after entry. But even if this use of “all” applies to all ‘humans’, Universalists are still forced to admit that in this text “those who believe” must necessarily be a subset of the “all people,” a group greater in number than “those who believe”. God as the ‘Savior of all’ can here be understood in a number of ways consistent with Scripture and this text, but it is impossible to legitimately use this verse to support the Universalism position. In fact, it argues against it. For within a Bible that affirms so strongly that not ‘all believe’, not only are Universalists unable to produce a single text that does affirms positively that ‘all believe’ (even if ‘in the end’), they are faced with this text, one of their founding pillars, which so clearly highlights that ‘those who believe’ (and consequently, enter life) are not ‘all people’ of whom God is the Savior.
In The Sovereignty of God (1928, Banner of Truth), A. W. Pink writes  briefly about a few passages which seem to teach strongly an ‘unlimited design in the death of Christ’ and demonstrates how contextually these references to ‘all’ cannot refer universally to everybody without exception:
[2 Corinthians 5:14]
In 2 Cor. 5:14 we read, “One died for all.” But that is not all this scripture affirms. If the entire verse and passage from which these words are quoted be carefully examined, it will be found that instead of teaching an unlimited atonement, it emphatically argues a limited design in the death of Christ. The whole verse reads, “For the love of Christ constraineth us; because we thus judge, that if One died for all, then were all dead.” It should be pointed out that in the Greek there is the definite article before the last “all,” and that the verb here is in the aortist tense, and therefore should read, “We thus judge: that if One died for all, then they all died.” The apostle is here drawing a conclusion, as is clear from the words, “we thus judge, that if… then were…” His meaning is, that those for whom the One died are regarded, judicially, as having died too. The next verse goes on to say, “And He died for all, that they which live should not henceforth live unto themselves, but unto Him which died for them, and rose again.: The One not only died but “rose again,” and so, too, did the “all” for whom He died, for it is here said they “live.” Those for whom a substitute acts are legally regarded as having acted themselves. In the sight of the law the substitute and those whom he represents are one. So it is in the sight of God. Christ was identified with His people and His people were identified with Him, hence when He died they died (judicially) and when He rose they rose also. But further we are told in this passage (ver. 17), that if any man be in Christ he is a new creation; he has received a new life in fact as well as in the sight of the law, hence the “all” for whom Christ died are here bidden to live henceforth no more unto themselves, “but unto Him which died for them, and rose again.” In other words, those who belong to this “all” for whom Christ died, are here exhorted to manifest practically in their daily lives what is true of them judicially: they are to “live unto Christ who died for them.” Thus the “One died for all” is defined for us. The “all” for which Christ died are the “they which live,” and which are here bidden to live “unto Him”…
[1 Timothy 2:5-6]
“For there is one God, and one Mediator, between God and men (not “man,” for this would have been a generic term and signified mankind. O the accuracy of Holy Writ!), the Man Christ Jesus; who gave Himself a ransom for all, to be testified in due time” (1 Tim. 2:5, 6). It is upon the words “who gave Himself a ransom for all” we would now comment. In Scripture the word “all” (as applied to humankind) is used in two senses—absolutely, and relatively. In some passages it means all without exception; in others it signifies all without distinction. Which of these meanings it bears in any particular passage must be determined by the context and decided by a comparison of parallel scriptures. That the word “all” is used in a relative and restricted sense, and in such case means all without disctinction and not all without exception, is clear from a number of scriptures, from which we select two or three as samples. “And there went out unto him all the land of Judea, and they of
Jerusalem, and were all baptized of him in the , confessing their sins.” (Mark 1:5). Does this mean that every man, woman and child from “all the land of Judea and they of river of Jordan Jerusalem” was baptized of John in ? Surely not. Luke 7:30 distinctly says, “But the Pharisees and lawyers rejected the counsel of God against themselves, being not baptized of him.” Then what does “all baptized of him” mean? We answer it does not mean all without exception, but all without distinction, that is, all classes and conditions of men. The same explanation applies to Luke 3:21. Again we read, “And early in the morning He came again into the Jordan , and all the people came unto Him; and He sat down, and taught them” (John 8:2); are we to understand this expression absolutely or relatively? Does “all the people” mean all without exception, or all without distinction, that is, all classes and conditions of people? Manifestly the latter; for the Temple Temple was not able to accommodate everybody that was in at this time, namely the Feast of Tabernacles. Again we read in Acts 22:15, “For thou (Paul) shalt be His witness unto all men of what thou hast seen and heard.” Surely “all men” here does not mean every member of the human race. Now we submit that the words “who gave Himself a ransom for all” in 1 Tim. 2:6 mean all without distinction, and not all without exception. He gave Himself a ransom for men of all nationalities, as we read in Rev. 5:9, “For Thou wast slain, and hast redeemed us to God by Thy blood out of every kindred, and tongue, and people, and nation.” That this is not an arbitrary definition of the “all” in our passage is clear from Matt. 20:28 where we read, “The Son of Man came not to be ministered unto, but to minister, and to give His life a ransom for many,” which limitation would be quite meaningless if He gave Himself a ransom for all without exception… Jerusalem
“But we see Jesus, who was made a little lower than the angels for the suffering of death, crowned with glory and honour; that He by the grace of God should taste death for every man” (Heb. 2:9). This passage need not detain us long. There is no word whatever in the Greek corresponding to “man” in our English version. In the Greek it is left in the abstract—“He tasted death for every.” The Revised Version has correctly omitted “man” from the text, but has wrongly inserted it in italics. Others suppose the word “thing” should be supplied—“He tasted death for every thing”—but this, too, we deem a mistake. It seems to us that the words which immediately follow explain our text: “For it became Him, for whom are all things, and by whom are all things, in bringing many sons unto glory, to make the captain of their salvation perfect through sufferings.” It is of “sons” the apostle is here writing, and we suggest an ellipsis of “son”—thus; “He tasted death for every”—and supply son in italics. Thus instead of teaching the unlimited design of Christ’s death, Heb. 2:9-10 is in perfect accord with the other scriptures we have quoted which set forth the restricted purpose in the Atonement; it was for the “sons” and not the human race our Lord “tasted death.”
 Pink, A. W. The Sovereignty of God: 1928, Banner of Truth, p. 64-67.