I say yes, and have reasons why.
Biking shoes (Mountain or road) have a very stiff sole, which directs all of the power from your foot into the pedal, giving you more power for the same effort, as well as tending to protect from tearing the tendons in the bottom of your foot. Add to that that the construction of cycling shoes helps to let heat escape, causing less sweating and therefore more comfort. The particular kind of biking shoe you need, however, depends entirely on the kind of riding you plan to do.
Performance road, or distance road: for this, you would need a road specific shoe as the sole is stiffer and the ventilation is better. They also tend to be lighter, and have more advanced fastening systems.
Performance mountain: For this you will need a mountain shoe, which will give you a walking surface, in case you need to carry your bike over obstacles that you cannot otherwise navigate.
Urban, cruising, touring: For this as well, I recommend a mountain shoe, generally. The primary reason is that they can be walked in fairly easily. This will help if you are spending time off of the bike after or in the middle of a ride. For touring, you can use a road shoe, bit stopping on the way is a bit more problematic in that instance.
If you are considering a clipless system, simply match the type of clipless system with the kind of cycling shoe you will be using (based on your riding needs)