As above, your engine will be running without filtered air. It is mostly ok, until something like a small stone/grit/leaf/fly/so forth... which then enteres the engine and either causes untold internal damage, jams something, or starts to gum things up such as valves, or start to foul sensors such as the MAF sensor.
you may see drag cars run like this, however engine rebuilds are usualy done by the 1/4 mile, other than that nearly all other race cars I know of run some sort of air filtration to ensure reliability.
It is best to replace the lid to the filkter box (as well as leaving said filter in place!). If you want the induction note on the cheap, just remove the cold air pipe from the box or drill the box on the "dirty air" side, it has the same effect.
In both cases, the engine will now draw in warm air which will reduce power slightly, how much you will loose I dont know. You will also increase fuel consumption, mostly because your foot will usualy be further into the floor for the noise!