Do items de-spawn in Diablo?
It depends on whether you play singleplayer or multiplayer.
In singleplayer, dropped items stay where they are until you defeat Diablo or start a new game with this character (this also applies to the entire dungeon, no levels will repopulate until you start a new game). You can save and load the game, exit and restart the game, doesn't matter, everything is saved. Once Diablo is defeated however, the open game of your character ends and items disappear when starting a new game. In singleplayer it's pretty common to keep a "stash" of gold, spellbooks and other useful items on the ground near the town portal area. Just remember to take it with you before you defeat Diablo or start a new game (don't worry, Diablo doesn't drop anything special in Diablo 1).
In multiplayer, it works as @Tom describes. A game stays open as long as at least one character in it. Once no players are left, the game is closed and disappears forever. In multiplayer, you usually use a mule character to serve as your stash. This does however require you to play with a friend you can trust not to steal your gold. Alternatively you could also try to keep the game open forever, but that risks a disconnect and closing of the game. Or you login to the same game from two different machines to handle the muling yourself.
It's been a number of years since I played Diablo, however, if I remember correctly, you can drop items in town during a game and they will stay there until the game ends. If you fail to pick up all of your items prior to exiting the game, you will lose anything on the ground at that time.
Just a couple months ago I played a whole bunch of Diablo I.
The ground is your stash
Yes, in singleplayer Diablo I, standard practice is to throw excess items on the ground. They will not despawn. The key is Diablo I uses "Save Game" and "Load Game" mechanics, so the game-state is 100% saved. (It's easy to forget this, since Diablo II and III made a point to disallow saves and provide a resurrect mechanic).
Gold only stacks to 5000 per space, so this soon becomes necessary. It's also wise to save back health and mana pots for the off-chance of encountering an Eldritch Shrine. (For mystery shrines, save game first, then reload and decide if you really want the effect. The shrine effect won't change when you reload).
You'll have quite a lot of stuff on the ground, between potions, marginal equipment with specialty resists or buffs (notably magic buffs), and a sea of gold laid out in 5x4 grids.
Taking it with you
Near endgame, you cull this down to the 40 spaces you can carry. Equipment is the hard choice. You're very limited as to how much gold you can carry, so bringing gold forward is a huge problem. You generally transfer that into skills.
There's very little chance of exchanging gold for desirable magic items. Griswold's items are of middling value, and Wirt's one item is good but usually not an upgrade for you -- the problem is, at endgame you have no ability to force their inventory to re-randomize - it remains the same when saving and reloading the game. So they are no help at all.
On the other hand, Adria the witch completely regenerates her list of spellbooks and elixirs with each save/load. So you burn gold at Adria; reloading and buying spellbooks (and if in Hellfire, elixirs which give +1 to a particular stat). Adria's selection is varied enough that you're sure to get something usable every 2-3 reloads.
For spellbooks, it's absolutely vital to keep a full set of +magic stat gear to the very end. You will need it to learn Adria's higher level spellbooks. Even silly spells like Town Portal or Healing are beneficial to skill up, because it reduces their mana cost.
You finish with all 40 slots filled with a careful set of choices and fairly little gold. Then you either head off to kill Diablo, or just start a new game. (Killing Diablo ends the game, so anything still on the ground is lost.)