Does this sentence mean you are not allowed to or you are not able to ?
If you mean "to be allowed" there's a completely different word. Using this sentence as an example, it would become: "Labrum implere tibi non licet", whereas "tibi non licet" means more or less "to you is not allowed".
You can read more here: https://en.m.wiktionary.org/wiki/licet#Latin
I confess I had a vague memory of the word, I had to do a quick research before I could write this comment. ^^" I knew it was a different term but I couldn't remember it.
Thanks for answering.
Why do they have different a sentence structure ? Would Labrum implere non licet and Labrum implere tibi non posse be right ?
Because licet is an impersonal verb (it only has the third person) that requires the dative case: mihi, tibi, ei, nobis, vobis, eis. Posse instead can and has to be declined: possum, potes, etc.
Labrum implere non licet means "It is not allowed to fill the tub". Labrum implere tibi non posse perhaps (take it with a grain of salt, I'll be waiting for either a confirmation or a correction by the moderators or some Latin teacher/professor) could mean something like "It is not possible to fill the tub for you" - meaning your tub in particular has some non specified issue, not that you can't fill it. The sentence in the exercise instead means exactly that you can't fill the tub, in the sense that you are not able.