The Crown Prince or Crown Princess does not have any authority over the Fire Nation itself.
However, they are active in war ventures, and are allowed to lead charges or attacks to opposing districts. Similarly, they can look over an invasion attempt, such as how Azula was a supervisor in the drilling of Ba Sing Se's walls.
In the Fire Lord's chamber, the Crown Prince is given the right hand spot from the Firelord, meaning that he is more influential than the high generals and high admirals. He has a say in the meetings, although the Fire Lord is still ultimately the figure in power.
Like other nobility the Crown Prince is given luxuries, an array of servants and a Royal Parade, if needed.