I like to do High Intensity Interval Training (HIIT), and each time I do it my lungs are burning by the end. Is it ok to do cardio of this intensity (and that's hard on the lungs) everyday, or do your lungs need 48 hours to heal like your muscles do when you're weight-lifting?

Also can I do the same kind of HIIT everyday (ie running several days in a row), or do my leg muscles need 48 hours to heal (like in weight-lifting) since they're being worked so hard each time?

no one knows?

Depends what your goal is. If your lungs are burning you are more likely to be building fitness and not losing fat but you can do it every day. Weights you should do alternative days

Basically speaking, the higher the intensity of exercise the longer the recovery period should be. You should not do HIIT every day - 2,3 times/week max.