Not unless it's only a few thou thick. Anodizing takes away some of the parent metal, but we're talking a maximum of 2 THOUSANTH of an inch. IF the coating is cracked, the cracks MIGHT act as stress risers on a parts that sees severe fatigue loads.
As far as temperature resistance goes, the only issue is that Al2O3's coefficient of expansion is ~1/5 of pure Al. This means if it gets too hot, or repeated heat cycling could cause the coating to crack. I do notice my IC end tanks look a little crazed, but it's only noticeable when viewed closely at certain view angles. You typically wouldn't want to go much more than 300C, and even then it'd have to be coated with that in mind.