See meanings 9 and 10 here.
What is "burning scenes" watching movies? Can't concentrate well on it or remember well?
Mel-Yeah, well, you're tired.Mel- I saw you sleeping.
Jim- I was burning certain scenes into my memory.
What is the situation? You haven't told us.
She "caught" him with his eyes closed while they were supposed to be watching a movie together. He makes a playful excuse that he wasn't napping he was just "burning" certain scenes into his memory. (By closing his eyes to allow the image to remain on his retinas and fresh on his mind.)