Is there a way I can use an iPhone that's logged onto someone else's iCloud?

I recently found an iPhone 5s out in the street perfectly fine but it's logged into the owners iCloud and in order for me to restart it I have to put in that persons password and I've been told that there's no way for me remove that account. So I was wondering if anyone knows if it's safe to use it like that and just add another account or if I can use it some other way?

Correct, as long as the phone is associated with an active account, and once it has been reported lost/stolen, you won't be able to use that phone, on any carrier. So no there's no way around it.

So turn the phone in at an Apple store or the carrier that appears on the SIM card (if it has one), or the carrier that shows up when the phone boots up. And they can get the phone to it's owner in less than a days time if they are local. Now you "MIGHT" luck out when returning the phone, and find an owner that has already replaced their phone, and is willing to do what it takes to unblacklist the phone, so you can use it. But that would be the rare person.

So don't be a D bag and try to keep the phone.

There's no way around that & that's by design.