Do Calvinists believe that God does not love the world, but only the elect? I seem to remember hearing somewhere that at least some Calvinists believe that, contrary to what seems to be a literal reading of John 3:16 and other verses, that God, in fact, does not love the world, but only the elect.

So, is there a certain group of Calvinists (denominations) that hold to this belief?  If so, which ones?  Also, what is the explanation for verses like John 3:16 that seem to indicate that God loves not only those who come to Him, but also those who reject them.


  “For God so loved the world, that he gave his only Son, that whoever
  believes in him should not perish but have eternal life." John 3:16
  ESV
  
  And Jesus, looking at him, loved him, and said to him, “You lack one
  thing: go, sell all that you have and give to the poor, and you will
  have treasure in heaven; and come, follow me.”  Disheartened by the
  saying, he went away sorrowful, for he had great possessions.  Mark
  10:21-22 -> The rich young ruler who doesn't seem to have been one of
  the elect.
  
  The Lord is not slow to fulfill his promise as some count slowness,
  but is patient toward you, not wishing that any should perish, but
  that all should reach repentance.  2 Peter 3:9 ESV
  
  And Jesus said, “Father, forgive them, for they know not what they do.”
  Luke 23:34 ESV Jesus asks for the forgiveness of the ones that were rejecting 
  Him and crucifying Him.
If any famous Calvinist could be accused of not believing God loved everyone it would be John Owen one of the foremost Calvinistic theologians. Owen (like Calvin as well) can be misunderstood as those 'who did not believe Christ died for all'. However the more I have read of their works the more I have become convinced that actually nobody has ever-believed that God does not love all men (that is nobody 'truly Calvinistic')

This is really confusion over terminology. It has to do with the way we think of the elect.  Since God is some sense sent his Son for the elect only, as they alone will benefit by his death, we must define 'love' in different senses.  God may love sinners who reject him, but in what sense is that love present when the sinner is bound under everlasting burnings? It is with this in mind that different Calvinists and in some way all Christians grapple with different meanings of the word love, and some accuse others of not believing in love based on the way they define this 'universal love'.  In my case I would simply say as a Calvinist, God loves all men equally but from His eternal planned salvation only loves the elect effectually. But I am not fixed on my current expression. Faith believes in God's love so does not desperately seek exact terms for everything but can repose and rest in faith.

In any case here is what Owen say's when accused on believing God did not love all: (All quotes from John Owen's Works, P555-557, CHAPTER XXXI Of election and universal grace—Of the resurrection of Christ from the dead)

He notes the following verses implies a general love for all in some sense: John 3:16, 17, 6:33, 4:42; 1 John 4:14; John 12:46, 47; Mark 16:15, 16; Col. 1:23, 28; 1 Tim. 2:1–4; 2 Pet. 3:9; 2 Cor. 5:19; 1 John 2:1, 2.

Then he says:


  That God is good to all men, and bountiful, being a wise, powerful, liberal provider for the works of his hands, in and by innumerable dispensations and various communications of his goodness to them, and may in that regard be said to have a universal love for them all, is granted; but that God loveth all and every man alike, with that eternal love which is the fountain of his giving Christ for them and to them, and all good things with him, is not in the least intimated by any of those places of Scripture where they are expressed for whom Christ died, as elsewhere hath been abundantly manifested.


Then he says that actually it is our faith in God's general love for all that that we are saved by not our faith in his love for the elect, because the Bible does not give us faith in who is exactly elect (not saying that after we are saved we do not have full assurance that we are inn deed elect):


  This is indeed the only true and solid ground of coming unto God by Christ, that God hath infallibly conjoined faith and salvation, so that whosoever believes shall be saved; neither doth the granting of the pretended universality of God’s love afford any other ground whatever; and this is not in the least shaken or impaired by the effectual love and purpose of God for the salvation of some...We say, though God hath chosen some only to salvation by Christ,—yet the names of those some are not expressed in Scripture, the doing whereof would have been destructive to the main end of the word, the nature of faith, and all the ordinances of the gospel,—yet God having declared that whosoever believeth shall be saved, there is sufficient ground for all and every man in the world to whom the gospel is preached to come to God by Christ, and other ground there is none, nor can be offered by the assertors of the pretended universality of God’s love. Nor is this proposition, “He that believeth shall be saved,” founded on the universality of love pleaded for, but on the sufficiency of the means for the accomplishment of what is therein asserted,—namely, the blood of Christ, who is believed on.


But after establishing God's love for all Owen reminds us of some facts about those predestined to glory which might lead some to the confusion:


  Some are chosen from eternity, and are under the purpose of God, as to the good mentioned. Those some are some only, not all; and therefore, as to the good intended, there is not a universal love in God as to the objects of it, but such a distinguishing one as is spoken against: Eph. 1:4, 5, “According as he hath chosen us in him before the foundation of the world, that we should be holy and without blame before him in love: having predestinated us to the adoption of children by Jesus Christ to himself, according to the good pleasure of his will.” Here are some chosen, and consequently an intention of God concerning them expressed, and this from eternity, or before the foundation of the world, and this to the good of holiness, adoption, salvation; and this is only of some, and not of all the world, as the whole tenor of the discourse, being referred to believers, doth abundantly manifest. 


He also reminds us that others are predestined to hell:


  Yet farther to evidence that this purpose of God or intention spoken of is peculiar and distinguishing, there is express mention of another sort of men who are not thus chosen, but lie under the purpose of God as to a contrary lot and condition: “The Lord hath made all things for himself; yea, even the wicked for the day of evil,” Prov. 16:4. They are persons “whose names are not written in the book of life of the Lamb,” Rev. 13:8; being “of old ordained to condemnation,” Jude 4; being as “natural brute beasts, made to be taken and destroyed,” 2 Pet. 2:12. And therefore the apostle distinguisheth all men into those who are “appointed to wrath,” and those who are “appointed to the obtaining of salvation by Jesus Christ,” 1 Thess. 5:9


Conclusion: Nobody really believes God does not love the whole human race:


  As the whole matter of our salvation must not be sought any where else than in Christ, so we must see whence Christ came to us, and why he was offered to be our Savior. Both points are distinctly stated to us: namely, that faith in Christ brings life to all, and that Christ brought life, because the Heavenly Father loves the human race, and wishes that they should not perish. (John Calvin on John 3:16)


Notes:

All of John Owen's works can be found here.
Here are a couple posts I have made before about this subject before but when I wrote them I had never found these little nuggets posted here:

&quot;God desires all men to be saved&quot; from a Calvinist perspective?

Does God hate unborn children?