Even if you do clutchless shifts the right way as Kevin explained, it will still greatly increase the wear on the transmission.
These are two gears from an old GSXR transmission. Note that this isn't the set for one gear. These are neighboring gears on one shaft, probably first and second from the input shaft.
The raised areas on the left gear are called dogs. When you shift from one gear to another, the dogs on one gear lock or disengage from the slots on the gear next to it, like you see in the gear on the right in this picture.
All the power your bike makes at any given moment is transmitted through those dogs to the slots, and then from the input gear to the output gear. You get a false neutral when the dogs slip out of the slots.
This can happen if you put more power through the gear than it was designed to take. Turbo bikes have big problems with popping out of gear when they get on the boost unless the faces of the dogs are cut back (undercut) to make sure they lock in. When you do a clutchless shift the dogs can pop out because you are applying power when the dogs are only partially engaged.
The problem is that even when the gear does fully engage, having a load on the dogs when they meet the slots will cause wear at the top of the dog. You can see this in the picture. The shiny, beveled part on the dog at the top of the gear isn't supposed to be there. That has been ground and hammered away because of shifting without the clutch. Whoever had this bike lost the ability to do clutchless shifts some time ago because the gear can't engage, and with half of the surface area on the face now missing I'll also bet it was popping out of gear when he got on the gas hard.
I've seen guys destroy their transmissions like this in just a couple thousand miles. If you do it the right way you can put off problems for ten thousand miles or more, which is a lifetime for a race bike, but you will be causing more wear than if you use your clutch all the time.
Because you have to remove the engine and split the cases to repair this, you can easily expect a four-figure bill to replace a couple of gears with worn dogs. I always told guys to use the clutch even on the track unless they were racing for money or if they had someone to pay their bills. Work on your corner speed instead of trying to save a millisecond by not using the clutch. And in my book you're a straight-up squid if you don't use the clutch on the street.