100% safe. The oxygen deprivation is relatively rapid and a small amount of gas. I can't imagine anything making it into the actual meat of the bird. However, even if it were to do so - the gases are harmless. This isn't 'gassing' with say, carbon monoxide (which becomes toxic in high concentrations (5% +) that poisons the animal, but essentially displacing oxygen with an inert gas (nitrogen, helium) to lose consciousness and asphyxiate.
In a perfect world, this causes little to no unnecessary distress to the bird, or at the very least, completes a relatively unpleasant task as humanely as possible. Either way, as long as you aren't poisoning your birds, they won't poison you on the table. I hope that offers some clarity (but if it rambles on - it's almost 1 am) then feel free to follow-up.
