Please discuss this issue with your physician. In general, if you are allowed to have clean water on your burn, it should be safe to immerse. As far as your concerns about pain – it is really your choice whether to wait until the burn has healed somewhat. It is legitimate and permissible to
delay mikveh if immersion would be very painful.
If you decide to immerse on time, then when you go that morning for the dressing change, ask them to instruct you how to re-bandage your hand in the evening and make sure you have the needed materials. You should arrange with the mikveh to be the first one to immerse that night (assuring that the water will be the cleanest). In this situation, a
single immersion in the mikveh is sufficient; you do not need to dunk multiple times. The bracha should be recited before you immerse.
Refuah shleimah.