Can you put a person's birthday under the Buddha's incense burner?

Yes, but it shouldn't have any effect, because bodhisattva doesn't need to look at personality. If you burn incense, he will bless you. You are good at doing good deeds, even if you don't burn incense, the Bodhisattva will bless you. It doesn't matter how much incense you burn if you feel sick after doing something bad.