you do burn fat in an anaerobic phase, but a smaller percentage of the total calories burned while anaerobic are fat calories than the percentage of fat calories burned while doing aerobic work.
the thing is, though, that you burn more total calories when you work anaerobically through intervals and other high intensity work, so you still might burn as much fat. the general understanding now is that weight loss really hinges on the calories you consume and the calories you burn off. if you're burning off a lot of calories by doing intense anaerobic work, you're still helping yourself lose weight because you lose weight when you consume fewer calories than you use.
also, anaerobic work increases your cardiovascular capacity and makes you more fit overall. the idea that one shouldn't go anaerobic and always stay in an aerobic zone is dated. however, if you're worried about it, you can try to get a mix of anaerobic and aerobic work to cover your bases.