I had a bad experience a few years back while flying with my backpacking stove. I was trying to check in for an internal flight, one of few in that trip. On the check-in counter, the security officer decided that I cannot send my Whisperlite international with my luggage. He was really stubborn and wouldn't compromise. It looked totally ridiculous to me for lots of reasons. First of all, I was flying with it (and with my gas stove) for years, and never encountered a problem. Second, I couldn't figure it out. I mean, people travel with stoves around the globe constantly, how can it be illegal? It ended with a lost stove and a forced visit to REI after I landed to get a new stove.
Well, it turns out that the security officer was partially right. The strict regulations forbid anything flammable aboard. So clearly no fuel is allowed and even no residue of fuel. But when you think of it, it might be problematic for anything that even SMELL flammable...
This is why, by the way, that the regulation is pretty vague and up to the specific security officer you might meet. I mean, my stove's sake depends on someone's smelling abilities?? Since it is so vague, it changes from airport to airport and I guess that it might even change from one security officer to the other. Luck is of the essence...
Anyhow, in order to make things simple, here are my recommendations to avoid unexpected problems at the airport.
I highly recommend flying without the canister. Just leave the half full one at home and get a new one after you land. This shouldn't be a problem throughout North America and Europe. In places you think you won't find a canister, I wouldn't go with this kind of stove from the first place.
As for the burner and the other stuff like igniter, cookware and so, it really shouldn't be a problem. Just keep it clean and send it with the rest of the checked luggage. Don't take it in your hand carry.
In most multi fuel stoves, you can disconnect the fuel bottle from the burner. The burner isn't the problem. Just clean it up a bit along with all the parts (except the bottle) and you're good.
As for the fuel bottle: empty it, wash it with warm water with a bit of soap, rinse and dry it. It should be OK. Pack it open.
In cases where the bottle is an integral part of the stove (Coleman 442 for instance) it's a bit harder but try and wash it too. The problem here is that you might lose the whole pack, whereas in cases where the bottle is an independent part, you can still keep the burner.
Tip: wash the pack sack with soap to take the smell away. You don't want to argue about it on the check-in counter.
Just keep the fuel bottle empty (Or the burner if it holds some fuel). It is pretty safe since alcohol vaporizes quickly and hardly leaves any smell.
Disclaimer: The information above is based on my experience and does not replace the official regulations. Ask your airline representative for their specific rules and regulations and act accordingly.