How long do dropped items persist?

It seems that there is a limit of 400 "loose" items at once, and if that limit is exceeded then the oldest item disappears. As of version 1.2, items of the same type will stack when they're near each other, thus greatly reducing the likelihood of this happening.

Also, quitting the game destroys all items on the ground.

http://terraria.gamepedia.com/Items#Despawning

There is no time limit. You can keep an item on the ground for as long as you want just as long as you don't turn off the server or exit the single player world. There is however a 200 item limit that can be on the ground.

As is stated on the wiki it was never confirmed but proven by using the chest duplication glitch. This is to prevent the world from being overwhelmed with unneeded cleaning and item lag (too many non-active tiles (items) on the floor can cause server/single player lag).

Coins count as 1 item even if it is in multiple. For example, when you pick up a bronze coin it might say 93 but on Terraria it will count as one item on the floor. In the condition of your gold coin, I'm sorry, I know the pain of losing gold coins (believe me I lost platinum before (100 gold coins coin tier)) but you probably logged out of your world and it disappeared, if not then you might have picked it up without noticing.

EDIT: As is now, there seems to be a 400 item limit, as stated on the wiki.