No, not at all, not while on the flight. They do pump in fresh air, although the cabin altitude will be higher than you're used to.
The only vague health problems I've had when flying were catching colds, and those were on long flights (think Honolulu - Baltimore nonstop) and in pre-mask days. I wouldn't think you'd have any problems, but keep your inhaler handy and make sure you've taken whatever meds you're on beforehand. The big thing is keeping hydrated. You'll dry out fast at altitude, so have a big ole water bottle handy and hit it frequently.
(The cabin air pressure is much, much higher than the outside air pressure when you're at cruising altitude, but it's still lower and drier than when you're on the ground.)