My g-g-g-uncle was a member of the Illinois infantry, and died in Tennessee in the early years of the war of "brain fever". His family were poor farmers. Is it likely he was buried near the camp, or would his remains have been sent home?
Only officers (and even then it depended on whether that man's family was wealthy, or if he had a wealthy friend[s] in-camp) would be shipped back home except in very rare cases. Depending on the campaign, and how connected that Union Army was to its supply line, a corpse might have been sent back along with the wounded to then be transported home for burial at the family's discretion. If this was done, they would have almost certainly been embalmed (pretty much the only way a corpse would make it home without rotting), yet this was a somewhat expensive procedure (between $25-$50, which represented as much as 2-4 times a soldier's monthly pay). Colonels and above did get sent home in this way with some amount of regularity in the Union, but for anyone ranked lower than this, it would have been somewhat rare.
So, if a person wanted their body shipped home, on top of the embalming costs, there was the practical matter of paying to get it shipped, which would not have been cheap. As a result, the vast majority of soldiers, on both sides of the ACW, were buried near camp, or on the battlefield itself. If, as you asked, a soldier died from illness in-camp or on the march, they would have been buried in a suitable location (higher ground, not near a water supply, etc.) by his comrades, usually with the help of the company chaplain or surgeon.
[Sources - Bruce Catton, 'Mr. Lincoln's Army'; James McPherson, 'Battle Cry of Freedom']