It is ambiguous, but we would probably assume that the word "his" is referring to the boss, for a number of reasons:
- Only you can see the fairy - they are a supernatural entity and it is unclear that you can touch them.
- Fairies are typically conceived as female (although they don't have to be). The stereotypical boss is probably still imagined as male (although they by no means have to be).
- Your boss is annoying, so there is some logic to shutting him up.
- It is counterintuitive that the fairy would ask you to glue their own lips together. As it is an unpleasant thing thing to do to someone, it is intuitively more likely that the fairy would ask you to do it to someone else.
- There is a sense that you are being asked to do something risky or transgressive, something that might get you into trouble but which carries the promise of a reward. This makes no sense if you are talking about doing something to a fairy that only you can see.
Of course, if someone were asking you this, it could be a trick question (possibly one designed to illustrate gender biases - although as I've said, gender isn't the only reason for interpreting the sentence this way).