The Apostle Paul expressed such a desire in Romans 9:
9 I speak the truth in Christ—I am not lying, my conscience confirms
it through the Holy Spirit— 2 I have great sorrow and unceasing
anguish in my heart. 3 For I could wish that I myself were cursed and
cut off from Christ for the sake of my people, those of my own race, 4
the people of Israel. Theirs is the adoption to sonship; theirs the
divine glory, the covenants, the receiving of the law, the temple
worship and the promises. 5 Theirs are the patriarchs, and from them
is traced the human ancestry of the Messiah, who is God over all,
forever praised! Amen.
In mentioning his desire, he does not expressly either affirm or deny the possibility of accepting extra suffering in order that others might be saved. However, he gives the impression that it is not possible, leaving all such decisions to God in His sovereignty. Voluntarily accepting suffering for people who are still alive is however beneficial:
15 But the Lord said to Ananias, “Go! This man is my chosen instrument
to proclaim my name to the Gentiles and their kings and to the people
of Israel. 16 I will show him how much he must suffer for my name.”
At the heart of the problem is the suitability of the sacrifice. Christ is sinless, therefore his sacrifice was acceptable to God to atone for the sins of humanity. Anyone other than Christ offers a lesser sacrifice that is likely inadequate to the task.
Another problem has to do with the purpose of suffering and the fate of the elect. If suffering falls short of eternal damnation, then it is for a purpose of God's, to perfect a person and prepare them for heaven. The work of God must be completed before they can be admitted. No imperfect person will enter heaven. Only excess suffering not needed to accomplish this perfection could be removed. If there is excess suffering beyond what is needed, then that exposes God to the charge of being unloving.
If the suffering is that of eternal damnation, then you have the puzzle of election, a sticky topic. No elect person can enter Hell or God's soverignty is overthrown, thus a saved person cannot trade places and enter Hell on another's behalf. Likewise, if a person is not elect, nothing you can do will change that fact without also making God out to be a liar. If a person's destiny is not certain, not a part of God's sovereign plan, then there is no such thing as election, and the Bible expresses an untruth; Jesus is made out to be a liar.
Sinning against an infinite God incurs an infinite cost.
St. Augustine of Hippo affirms a form of election:
Listen, thou ungrateful one, listen: “Ye have not chosen me, but I
have chosen you.” Not that thou mayest say, I am chosen because I
already believed. For if thou wert believing in Him, then hadst thou
already chosen Him. But listen: “Ye have not chosen me.” Not that thou
mayest say, Before I believed I was already doing good works, and
therefore was I chosen. For what good work can be prior to faith, when
the apostle says, “Whatsoever is not of faith is sin”? What, then,
are we to say on hearing such words, “Ye have not chosen me,” but that
we were evil, and were chosen in order that we might be good through
the grace of Him who chose us? For it is not by grace, if merit
preceded: but it is of grace: and therefore that grace did not find,
but effected the merit.
Augustine of Hippo. “Lectures or Tractates on the Gospel According to St. John.” St. Augustin: Homilies on the Gospel of John, Homilies on the First Epistle of John, Soliloquies. Ed. Philip Schaff. Trans. John Gibb and James Innes. Vol. 7. New York: Christian Literature Company, 1888. 353.