Smoking involves two physical processes, combustion and distillation. When you light your pipe, hot gases are produced (combustion) which, when you draw them through the tobacco in the bowl, vaporize volatile, aromatic chemicals in the tobacco (distillation). These are then drawn through the stem and into your mouth and are what you taste. The taste depends on what chemicals are in the smoke and on their quantity. This, in turn, depends not only on the type of tobacco (Virginia, burley, English, etc.) but also on the temperature of the smoke and the amount of the tobacco through which it is drawn. A hotter burning pipe will vaporize more chemicals and less volatile ones than than a cooler burning one and, the more tobacco the hot gases are drawn through, the greater the amount of chemicals that will be vaporized. Since the different chemicals in the tobacco have different flavors, differences in the temperature of the smoke will affect the flavor.
The shape and size of the pipe affects this process in two ways. First, all things being equal (mainly, how hard and fast you draw on the pipe), a narrow bowl will burn hotter because the gases will be moving faster. Second, a deep bowl allows the hot gases to be drawn through more tobacco, which releases more volatile compounds into the smoke. So we can expect tobacco smoked in deep, narrow bowls to taste different than the same tobacco smoked in a wide, shallow bowl.
Of course, a lot of other things affect the smoke, such as how hard and frequently you draw on your pipe (more making for a hotter burn and less making for a cooler one), the length of the stem (a longer stem cools the smoke, which will affect the taste as some of the chemicals in the smoke condense out of it), the thickness of the bowl (thicker means a cooler smoke), etc. Also important is the smoker's ability to taste things. Some people have a very sensitive sense of taste (like wine sommeliers) and others not so much.
So, the shape and size of a pipe bowl can affect the taste of the tobacco. A lot of smokers prefer to smoke different tobacco in differently shaped pipes for this reason. I prefer to smoke Virginias in deep bowl pipes because I think it makes them taste sweeter, and I prefer to English style tobaccos in wide bowl pipes because I think it enhances the smokiness of the Latakia.
Of course, each of us gets to decide for himself which pipe/tobacco combination tastes best, or whether it makes a difference at all.