Differences Between Smoothed DC & Not Smoothed DC With Respect to Heating
Adding a large capacitor will increase the heating of the bridge, assuming a constant resistance load. The average and the RMS currents through the bridge will both increase, so more heat (but also more output power). 

Because the capacitor charges in sharp pulses, the heating will actually be more than you might expect from the average output current increase. That is because the diode forward voltage increases with current, and a portion is actually resistive so some component of the heating is related to the current squared and not just the current (hence RMS current).