How long does money stay after death?
Items dropped in Terraria Single-Player Mode do not disappear until picked up, destroyed, or until the game has been saved and exited. Coins, despite their monetary significance, are treated just the same as any other item in-game.
Items can only be destroyed (currently) by full immersion in Lava or Magma. Thus, as long as you didn't burn up in a magma pool, chances are good that your coins are still waiting there for you, even if it takes a bit of work to find.
I'm unsure how items are handled on multi-player servers.
Items dropped in Terraria Single-Player Mode do not disappear until picked up
I don't fully agree. I've seen that untaken blocks of dirt, stone etc. disappeared after some time. I think this behaviour is related to count of objects of same type on screen. But I didn't notice that money disappear. It looks like that they stay until you pick up them :-)
PS: It looks like I found the maximal count of objects of one type (NPCs, chests, collectable items and so on) in game. It may be 0x3e8 magic number or 1000 in decimal representation. It's very easy to confirm it. Take more than 4 stacks of blocks and begin to throw them away one by one. After that try to collect them back. If number of blocks will be lesser then before it means that items disappear.
Items that are dropped on the floor stay there forever until you pick them up again. There are only a handful of exceptions to this rule:
Unless you're playing online (on a server), all items will disappear when you exit a world. Items will also disappear if the server restarts. Source
If there are more than 400 "loose" items in the world (not just on your screen - the whole world), the oldest items will start disappearing as more drops appear. Source
In Expert Mode (not to be confused with Hardmode), monsters will pick up any loose coins on the floor. The money can be retrieved by killing the monster that picked it up. Monsters that have picked up money will sparkle in with the color of the highest value coin they possess. If a monster picks up the money you dropped when you died, chances are pretty high that money is gone for good, since the monster will usually despawn once you revive, taking the money with it. Source
Items can be destroyed in lava if they have a rarity of white or gray (with a couple of exceptions). Any item with a higher rarity than that is immune to being destroyed by lava. Money is white rarity, so it can be destroyed. Source