A patricularly widely disseminated Biblical verse, of course: for God so loved the world, that he gave his only begotten Son. Perhaps its popular appeal has to do with its ease of emotional translation. Anybody who has a son can think: 'I know how intensely I love my son; but God loved the world even more than that!' But that process of emotional translation is problematic, too: which is to say -- once you move beyond the immediate emotion, it gets gnarly. What would be an earthly comparison? Is it: 'I love my son intensely, but I am happy to see him join the army and go to fight in a war, evem though he may well die, because I love my country more' ...? But that has an unpleasantly fanatical, nationalist edge (who could really love their country more than their children? Does't saying so really just another way of saying that you don't love your kids enough?) We might object that the parent in this scenario is faced only with the possibility of their son's death, where God sent Jesus into the world knowing that he would die: so God's sacrifice, and love, was greater. But that thought also branches in weird, and unappealing directions -- as if the love referred to in J316 is that of the parent of a kamikaze pilot, for instance. Or, worse, theologically speaking: God knows his son is going to his death; but he also knows, with an absolute divine certainty, that he will subsequently be reunited with him. The parent of a kamikaze pilot may or may not hope for that, but s/he cannot know it. Which leads to the unpalatable implication that a parent who gladly sends their child off to fly a plane into the USS Bunker Hill (or the Twin Towers) has a greater love than God's. Hard to swallow that notion.
Or to take it a step further. Would God's love for the world have been proportionately greater if the incarnation and atonement had involved not the crucification and resurrection of his son, but his complete annihilation? It seems to me hard to imagine it would; yet this is, perhaps, the implication of the John 3:16 logic. I suppose the issue here is the emotional logic that you demonstrate, or actualise, love by letting go of things, or losing them. Isn't the reverse closer to the truth?
[Later] Thinking more about this. I suppose the obvious objection to what I say here is that it misreads John 3:16 to interpret it in terms of loss. It is, rather, a verse that stresses what was giving, freely: the gift. But the gift -- the whole of the New Testament makes clear -- was death; a special death; a unique, magical death, the death of a god. But death is not a gift.