Seattle used to have this for an area of the city. It was called the "Free Ride Zone" - and it had some unexpected consequences. The primary problem is that homeless people would get on the bus because it's warm, and would fall asleep. The busses became rolling homeless shelters. This caused other people to take the bus less, because of public safety concerns, as well as simple quality-of-life issues like unpleasant…
The worst part in my view about the free ride zone was when you would take a bus from downtown to outside the zone. You'd need to pay when you left. Which was hard to enforce, so I think a lot of people walked off. Even if you wanted to play it right, you needed to remember when to pay and when not. I remember hearing from drivers that they didn't care if you pay anyway.
Of course in recent years in towns like Seattle or SF there is always somebody trying to say that the homeless are responsible for the downfall of everything.