I was in Myanmar last month. From the inside the official word was "No Way" to enter from Thai and go across to India, or the other way. One Italian guy spent a lot to time in Mai Sai getting "Official Permission" to enter, then got stopped about 100 k in where he had to pitch his bike on a truck and fly himself over the Closed Area. Same at the next Closed Area. Foreigners are not allowed to transit these closed areas, although their bikes (like his) can go through on a truck. I really do not read that as riding across Burma, but that was the rumor he had spread.... but he was riding airplanes and his bike riding trucks.
Once I got out of Yangoon (25 miles by truck) I was pretty much free to go where I wanted (Yangoon Area is "closed" to motorcycles).
I wandered down to the shore, only had one passport check. Was on a 250 Honda XR. Out of Mandalay I went east and north, but encountered closed areas further north, and could not push through to Mai Sai.
If you are not worried about time, money and frustration, go ahead and try, but common sense and experience tells me to skip it as an overland effort.
As for "legally" I would say there were probably numerous Brits running around/in/across Burma legally on some of the many BSA's they brought in when they controlled the country. That was probably before anyone really cared about writing about it.
I agree with David re: tours, probably more fishing or speculation than fact. There is one lengthwise bicycle tour that comes down to Yangoon from China, escorted by police on big bikes (older airhead BMWs in the photos I was shown).
Dr. G