Many Guild Wars 2 hearts ask you to collect or retrieve things. Somewhat breaking the logic of the hearts, you do not need to do anything with those. Picking them up gives you credit. You get more credit if you return them, but by the time you carve your way out of the bandit caves, your bandit-slaying has probably filled the heart already.
The game mechanics work just fine, but the logic is sometimes broken. The idea is that, instead of Innkeeper Bob giving me a quest to kill ten rats when I just slaughtered an entire nest of them, he will recognize that I am the rat slayer he has heard so much about, thanks, have a bit of silver and good karma. But I have completed hearts without ever seeing the heart-bearer, and he still sent me a nice “thank you” letter. I might have received most of my heart credit from checking his traps or collecting trinkets for him, but I never gave them to Bob. One of these is by the human starting area in Queensdale, a whole line of crawfish traps you can check, and then he sends you a letter thanking you without the intermediate step of “hey, what about those crawfish?”
Now you have a small stack of dredge ore, shiny red apples, jotun runestones, or broken sprinkler parts. Their only use is to turn in for heart or event progress. If those are for an event, great! Some of those events repeat rather quickly, so you could get another gold medal within the hour. If those are for a heart, you delete them. They are now completely useless and probably soulbound. Sometimes, you can still turn them in … for nothing. Hey, you already got your karma and thanks, stop being greedy. Other times, the heart-bearer will not even take them from you.
At least they usually come with a name attached so you know who to check. This is especially helpful with all the varieties of dredge ore that exist across the zones.