...it really matter how technical you want to be. I am aware of a few creatures (gate archon) in the bestiaries that wear medium/heavy armor and appear to use winged flight. If you are just referencing the rules. It only effects "barding" and "mounts". And the rules in encumbrance only counts only as "armor" not barding so has no relevance. And over 20 years of errata/faqs neither 3.x nor Pathfinder have really seen fit to clarify these rules to my knowledge. So by a RAW standard yeah an animal can fly just fine wearing heavy armor and/or carrying a heavy load.
But from my perspective of RAI. If something is wearing enough armor/carrying so much that its movement speed is reduced. Then winged flight should be restricted. The vast majority of winged creatures seem to hold to this, having only light or no physical armor. (Even those utilizing mechanical weapons.)
I'd expect to encounter table variation and want to clear a heavily armored flying companion with my GM before investing too much in the concept.