When a gas is compressed its temperature increases. When the compression is released, the temperature drops. This is the primary basis on how liquid nitrogen is formed. They take air and put it through numerous compression/decompression cycles. Each time they cycle it, the temperature of the air drops until it becomes a liquid. With the bicycle pump, you are compressing the air as you're pumping it into the tire so the pump is exposed to that increase in temperature. The air doesn't get decompressed in the pump, however, so it doesn't experience the cooling effect. (Since when you draw the pump back you're just taking in more air from the outside and not decompressing the air that was in there). So over time, this increased temperature causes the pump to become hotter.
The expansion/contraction of a gas under pressure is an adiabatic process. That means that as the volume changes, the temperature changes as well. In the bike pump the pressure of the gas is increasing so the temperature is increasing as well. You can feel that temperature increase in the pump, but if you were to measure the temperature of the tire you would see that it was a bit lower since the expanding gas was cooling in temperature.