I know that hurricanes can basically "burn" themselves out of energy if they get too strong, but can the same thing happen with a tornado?
Neither hurricanes nor tornadoes burn out because they get "too strong". Their strength is dependent on how supportive the atmospheric conditions are. If a hurricane stays in the same place for too long, moves over land, or moves over colder water, it will weaken. Likewise, a tornado will weaken if its parent storm moves into less favorable conditions and weakens.
All storms operate at "maximum strength" - they always are using all the energy that is available to them.