Well natural gas isn't safe for ANYONE at certain levels, including humans, dogs, cats, birds, etc etc -- but a natural gas leak of any kind WILL cause problems and quickly, death, in birds due to their smaller size and sensitivity to those sorts of agents. (Take teflon, for example -- the amount of gas produced when a teflon pan heats up is not anywhere NEAR fatal to humans, but kills a bird.) A similar example is the use of "miner's canaries" in coal mining many years ago -- a canary would stop singing prior to death when exposed to methane, carbon monoxide, etc. underground, which miners used as a way to keep from over-exposure themselves and avoid mining fatalities. The natural gas we use to heat our homes is partially composed of methane, so a natural gas leak in a home would likely have the same effect.
Propane, I'm not so sure about. Personally I would not specifically AVOID having a bird in a house with a propane fireplace any more than I would avoid having birds in my home, which uses a methane/natural gas stove with a pilot light. In MY PERSONAL OPINION, the likelihood of a gas leak from my stove (or fireplace) is so low and would so likely kill me, the cat, etc as well that it isn't something I am paranoid about (unlike the use of air fresheners and teflon pans when the kitchen doors are open). We can't possibly avoid EVERY SINGLE BIRDIE HAZARD EVER unless we all had gadzillions of dollars; the main thing I think your friend should consider is how far he can place the bird's cage from that fireplace, how likely a leak is, etc. If it is a situation in which a leak is highly likely and/or the cage would necessarily sit next to the fireplace, I might reconsider. Otherwise, if he is willing to offer a bird a loving home and that is the only major hazard and steps can be taken to mitigate possible danger, I would not say that he shouldn't consider adopting a bird just because of a fireplace.
My theory has always been to be as rational about fears of causing harm to my pets as possible. I don't use chemical cleaners, I don't cook with teflon pans when I can avoid it, I don't kill pests in my house with poisons. But beyond that, I try not to panic about potential hazards...because if I did, I'd never sleep again!