No, Do Not put the bit in till it bottoms!
Most router collets work by tightening a collet nut, which forces the collet into the tapered end of the spindle, causing the collet to contract around the shank of the bit. If you push the bit all the way into the tapered shaft, the collet isn't able to retract as deeply into the taper because the bit shank can't move any deeper. The bit may still be able to be tightened, but not as much as if the bit hadn't been bottomed. The same would be true for die grinders , etc., that also use collets, as well as for collets that have the hex milled directly onto the end of the collet and don't use collet nuts.
Drill chucks work differently. With drill chucks, the jaws of the chuck usually move outward as you tighten the chuck. If you bottom the bit, when you tighten the chuck, there will be a small amount of space between the bit and the internal chuck bottom, once the chuck is tightened.
If you're applying a high amount of pressure on the bit while drilling, or using the chuck in a hammering or impact mode, bottoming the bit might be a bad idea though. The high pressure or hammering action could cause the bit to slip deeper into the chuck and bottom, if the bit isn't held securely in place for some reason. The bottomed bit then might prevent the chuck jaws from easily retracting when you try to release the chuck jaws making it problematic to remove or change the bit.