My D&D character is a rogue, and I use shurikens. But I am uncertain whether I can pick them up after I throw them. Can I? And if it is possible, what are the rules for it?
Apparently not. It would seem that rules-wise, once you're finished attacking with that shuriken, it's vanished and the system no longer cares about it. The rules have no mention of ever picking up or recovering mundane thrown weapons — magical weapons return to your hand after they're thrown, but I presume we're just discussing the mundane shurikens from the PHB here.
From another perspective though: the rules also don't say they get used up when you throw them, so it could just be assumed you're picking them back up later. It's likely the rules didn't attempt to specify this because it would be up to the common sense of each group to handle it.
A lot of groups don't consider it worthwhile tracking arrows and just assume you have plenty (if your DM doesn't just hand you an Endless Quiver at level 5). Shurikens being almost as cheap, they might not be worth tracking either, or it might be worth ruling you simply recover your shurikens after combat.
There aren't any written rules that I could find either. There's nothing to prevent you from saying "after the battle, I collect my shurikens" as long as your DM permits.
Realistically it's a metal weapon, and just like swords and maces, daggers, etc., it doesn't exactly become 'unusable' once used. There are a few magic items I'd lean towards as well. I think there are gloves that allow you to draw a light blade thrown weapon as a free, and it ceases to exist after you've used it, but you can keep reusing that power.
If it's a magic shuriken, it'll automatically come back to your hand at the end of your turn.
During battle, however, I'd only ever try and retrieve them from dead enemies, because picking a shuriken out of a living enemy may lead to your DM houseruling some nasty surprises.
I never ran into a problem of collecting shurikens or arrows after a battle. Some of the arrows might be damaged and we rarely recovered all of them.
Shurikens I only ran into from a trap which fired a dozen of these from hidden holes when someone stepped on a pressure plate. It automatically reloaded so we triggered that pressure plate with a pole until it ran out of ammunition and then surprised the DM by picking up all that nice metal and lugging it home to sell. (We were pretty cash starved in that adventure and about half the money we gained (party of 6, lvl 3) was from two dozen shurikens. Most of the rest was from realizing Obsidian stone could be worth something when the DM described our arrival on a vulcanic Island with a beach of sharp obsidian).
Keeping in-line with the ruling on the arrows and assuming a shuriken to be a bit more sturdy I would allow players to collect them after a battle. A few would be lost. A few would be bent from hitting something at a strange angle. But most of them should be fine if you pick them up after the battle and optionally spent part of your evening resharpening them.