Eyelids might burn from an irritant dermatitis
from some exogenous chemical.
The irritant could be your eyeliner, mascara or eyeshadow. It also could be a moisturizer that gets on your fingertips or nail polish, nail harderner or cuticle treatment as your finger contacts your eyelids often and is proven to be a cause of eyelid dermatitis frequently. Skin testing can be done, see a dermatologist.