I have a being with a name. I capitalize its name when I mention it. But mostly, I refer to it as "it". I use "it" a lot in dialogue, as characters talk about this being. Should I capitalize "it", in these cases?
Capitalizing a pronoun gives it proper noun status or deity status. So writing It means either that the thing's name is literally It, like Stephen King's monster, or you're referring to a god the way the Abrahamic religions refer to Yahweh as Him even in the middle of a sentence.
Therefore, if the creature has an actual name, like Gormenghast or Voldemort or Donnie, use either the name or the third-person-neutral pronoun it, lowercase. If the creature had a gender, you'd have no problem referring to he or she, right? So it is no different.
This depends. If you are speaking about the creature as a generic item in a sentence:
"I watched as it ordered a Happy Meal," my friend told me.
then you would keep it lowercase:
If, though, "It" was an alternative name for the creature, you would capitalize:
Godzilla was so disgusting that we felt that referring to it by name would give it undue humanity, so we decided to simply refer to it as "It" from now on.
Some weeks later, my friend told me, "I watched as It ordered a Happy Meal."
I agree with Loren Ipsum about It as a personal pronoun, but I did have a little more to add. If you are looking for a way to keep dialogue and other description from being confusing, I would recommend a judicious use of italics. This wouldn't be necessary all of the time, but only so the readers would know the difference between it, the monster and it/its/it's.