Here is the commentary of Albert Barnes on this verse; I hope it helps.
1 Timothy 4:10
For therefore we both labour and suffer reproach - In making this truth known, that all might be saved, or that salvation was offered to all. The “labor” was chiefly experienced in carrying this intelligence abroad among the Gentiles; the “reproach” arose chiefly from the Jews for doing it.
Because we trust in the living God - This does not mean, as our translation would seem to imply, that he labored and suffered “because” he confided in God, or that this was the “reason” of his sufferings, but rather that this trust in the living God was his “support” in these labors and trials. “We labor and suffer reproach, for we have hope in God. Through him we look for salvation. We believe that he has made this known to people, and believing this, we labor earnestly to make it known, even though it be attended with reproaches.” The sentiment is, that the belief that God has revealed a plan of salvation for all people, and invites all people to be saved, will make his friends willing to “labor” to make this known, though it be attended with reproaches.
Who is the Saviour of all men - This must be understood as denoting that he is the Saviour of all people in some sense which differs from what is immediately affirmed - “especially of those that believe.” There is something pertaining to “them” in regard to salvation which does not pertain to “all men.” It cannot mean that he brings all people to heaven, “especially” those who believe - for this would be nonsense. And if he brings all people actually to heaven, how can it be “especially” true that he does this in regard to those who believe? Does it mean that he saves others “without” believing? But this would be contrary to the uniform doctrine of the Scriptures; see
Mar 16:16. When, therefore, it is said that he “is the Saviour of ‘all’ people, ‘especially’ of those who believe,” it must mean that there is a sense in which it is true that he may be called the Saviour of all people, while, at the same time, it is “actually” true that those only are saved who believe. This may be true in two respects:
(1) As he is the “Preserver” of people
Job 7:20, for in this sense he may be said to “save” them from famine, and war, and peril - keeping them from day to day; compare
Psa 107:28;
(2) As he has “provided” salvation for all people. He is thus their Saviour - and may be called the common Saviour of all; that is, he has confined the offer of salvation to no one class of people; he has not limited the atonement to one division of the human race; and he actually saves all who are willing to be saved by him.
(See supplementary note on
2Co 5:21. This passage however is not regarded a proof text now on the extent of the atonement, as the fair rendering of σωτήρ sōtēr is “Preserver.” Dr. Wardlaw has accordingly excluded it in his recent work.)
Specially of those that believe - This is evidently designed to limit the previous remark. If it had been left there, it might have been inferred that he would “actually save” all people. But the apostle held no such doctrine, and he here teaches that salvation is “actually” limited to those who believe. This is the speciality or the uniqueness in the salvation of those who actually reach heaven, that they are “believers;” see the notes on
Mar 16:16. All people, therefore, do not enter heaven, unless all people have faith. But is this so? What evidence is there that the great mass of mankind die believing on the Son of God?