Like others, if it seems reasonable to, I will leave what I found on trail for the person who lost it.
I experienced a weird 6-month period in which I found over $150 in cash on trail. This is over several trips on different trails. When it comes to cash I figure if somebody asks me if I found any cash on trail, I would definitely hand it over. However, I'm not going to go around asking every hiker I encounter if they did.
One time I found a new leatherman (the type that has about 1,000 attachments and seems great, but I never end up using), and since I hike at a pretty good clip, I asked every hiker I passed on trail if they had lost anything. A few people said no, and I came to an older gentleman hiking solo and when I asked him if he lost anything, he searched his bag and realized he was missing his tool, so we knew it was his.
Probably the best experience I had was a hike in Laos. While hiking some hills outside of Luang Prabang, I found a Korean passport on the trail. When I got back to town 2 days later, with nothing planned that day, I decided to visit every hostel in town (it's not a particularly big town) and ask around. Eventually, after almost giving up and deciding the guy must have gone back to the capital (Vientianne), I talked to a hostel owner and she confirmed her guest had lost his passport. The guy was so stoked when I handed it over to him. He didn't speak much English, but kept saying 'thank you.'
To this day I'm still not sure how he found out where I was staying, but the next day I came downstairs to the lobby and the lady at the desk told me I had a package. The guy had left me a bag of muffins, cookies, and bagels from the awesome French bakery down the street. That story still counts as one of my favorite travel experiences.
As for trash, if I can possibly pack it out, I do.