I lived and worked in Burma for a year in 1991, I was there as an aircraft pilot.
You can buy bikes locally, but they are all little ones (50 to 100cc), it was very rare at that time to see a bike bigger than 250cc.
We (the group of guys I lived with) had a little 50cc scooter we used just to get around our neighbourhood - we didn't try to register it to ourselves, we had the houseboy purchase it for us. The police (who are everywhere) kept a pretty close eye on us - they didn't care if we rode around within a mile or two of our house, but they always knew about it if we went any further. As expats, we were very much discouraged from driving cars or LandCruisers ourselves - each vehicle had a driver, and when we wanted to go somewhere, the driver took us there. The only exception was WAY upcountry (far north of Mandalay), where the LandCruiser couldn't go out of the village anyway, in that case, no problems with us driving it in the village.
AFIK about the present situation, the country is simply not encouraging (read: actively discouraging) independent tourism. I don't think it would be a good idea to try and go up against that government. Many others have tried and failed. They have no sense of humour.