I work out throughout the day. In the morning, I do 4-sets of pushups, my main workout after lunch, and I was thinking about adding calistenics at night. What I would do would be 2-set of pull ups (One for biceps and one for back) and 1-set of dips. I figured this would help my metabolism while I sleep. Is this a good idea or is it too much and even damaging?

I think you'll get better results, if elevating your BMR and RMR is a goal, by focusing more on your nutrition and your lifting routine.