No you should not be out of pocket. But if you fail to deliver the item to a police station then you are potentially liable to be charged with theft by finding. That's part if your contract with your customer.
As a cabbie, I committed many acts of theft by finding. After every shift, I'd lever up the back seat and there'd often be a few coins underneath. If I got to drive a replacement cab while mine was off the road, I'd be likely to find twenty or thirty dollars there.
Drunks were useful sources of supplementary income. There'd often be a bit of a rattle as they got out. If they had been good passengers, I'd call out for them to check the seat. If they had been cough - and there's a few people who are rude and obnoxious when drunk - I'd say nothing, just drive around the corner and scoop up the loose change before the next passenger did it for me.
For phones and things, there were a few occasions. If I remembered where the destination was, I'd go back and deliver it. Often I'd get a tip, but if not, no problem. If the passenger had flown off or whatever, the taxi company had a slot at the office for lost property. Just drop it in with a note, or ring up the next day to give details.
But by far the best solution was for me to glance over the passenger seat(s) after they got out. Unless the item had fallen down the side out of my view, I'd likely spot it and be able to chase after the passengers. A few phones and keyrings reunited that way.
As for theft by finding, yeah, maybe. But any cabbie has a ready made excuse. The next passenger must have knocked it off.
Uber, by the way, has a facility for the driver to take a picture of an item and send it to the passenger. And for the passenger to get in touch with the driver. That can fix a few problems right there. Unless the pax has left his phone in the cab...