The Burmese call this the cold season, but you'll still find yourself plenty toasty, with temperatures in the hotter areas hovering in the mid-90s°F (mid-30s°C) in October, down to the mid-80s°F (around 26°C) in February. Hotels fill up quickly and rates will be at their highest. Airline prices won't skyrocket but planes are quite small and book up quickly.
Southwestern monsoons, particularly from July to September, leave the country drenched, particularly in the western Rakhine State. The problem is not so much the rain itself as the aftermath, when dirt roads become impassible fields of mud. This is certainly the least expensive time to go. Around Pyay and Mandalay, you should be able to sightsee without getting soaked.
It's always hot and humid in Myanmar, but these are the months during which you'll roast if not careful. Temperatures in Yangon can easily top 100°F (38°F), with temperatures going even higher in Bagan and Mandalay. Still, because it's dry, hotel rates are not at their lowest, especially those with pools.