Are fins necessary/required for cooling in a closed room
There are two power functions you need to be concerned with: instantaneous power dissipation, and long term power dissipation. A large thermal mass (finned or not) will prevent your IC overheating due to a quick spike of power. Over the longer term, the heat needs to be dissipated to the outside environment. Fins are only effective if there is some airflow to remove the heat - this could be by simple convection or by forced flow (fan). In a sealed case, convection will be reduced compared to free air and a fan will also be much less effective.
For your case, if you don't get sufficient cooling from just normal internal convection, I would have a metal block from the top of the heat dissipating commponents to the case itself. That way, you increase the thermal conductance to the case. If you still have a problem with heating, then you could put fins on the outside of the case, which is presumably in free air.