If eardrum vibrates when hit, why don't we hear the air in the room that is continuously moving?

The answer depends on the kind of motion you are thinking of.

As PhysicsDoc points out, random motion at the molecular level averages out such that only a net effect will actually move the eardrum.

For larger scale movement, we need to understand a bit about how hearing works.

Perception of sound depends upon the movement of tiny hairs in the cochlea, which functions as a sort of mechanical frequency analyzer. Sounds transmitted into the cochlea excite the movement of hairs according to the frequency of the sound, which in turn stimulate associated nerves, creating the perception of sound at that frequency. See this for a more in-depth explanation. Due to this arrangement, it is not possible to "hear" a simple change in pressure – it must be periodic – a vibration, and it must have a frequency within some range (normal human hearing is between about 16 Hz and 20 000 Hz).

So, simple movement of air won't be detected by the ear as sound; there must be some repetition at some frequency.

Because of the very large number of gas molecules the collision rate is very high and so individual collisions cannot be detected. The impulsive forces simply integrate up to produce a steady pressure.

The ear drum mainly responds to change in external pressure ie.vibrations. When you are in a room , the ear drum is in equilibrium with the external pressure , hence it doesn't detect any vibrations . However if you start running or in a vehicle going at a high speed you'll will hear a faint sound of the wind, which is because of the fact that the air pressure on the ear drum is changing and not constant hence causing ear drum to vibrate.