you can get away with a character saying it once or twice. but I wouldn't use it often. Kind of a dated word.
Just found out I'm apparently old ... Am I a geezer myself?
Anyway, to answer the actual question: I'll out myself and say that I regularly use it. I have a character in my main story that is usually referred to as "old geezer" (because just being a geezer isn't enough apparently but, well, you see, he's really ancient considering that the person saying it is already kind of old) by his rather impolite stepson (?). I feel like I can get away with it though because it started out as a light-hearted comedy and the one using the word is the same person who keeps referring to the father of the person he likes as "father-in-law" upon first meeting. So clearly, he just doesn't care. And he is kind of crazy ...
Outside of that very specific case though? Nah. It usually doesn't really fit as just a descriptor of a person that I could throw into the narrative so dialog would be the only place where it is appropriate and most characters indeed wouldn't use it.