Shouldn't this be either "I do not celebrate my birthday" or "I do not celebrate the birthday of..."?
The first suggestion, yes, but the second, no, since a specific birthday is not given. I think another alternative that would be correct is "I do not celebrate birthdays".
This sentence is wrong in both languages. Who talks like that?? You have to have "my" in there or "the birthday of.." and in Romanian it has to be "Eu nu-mi sarbatoresc..." or "Eu nu sarbatoresc ziua (cuiva)".
Both sentences are so wrong. Who talks like that? You have to say whose birthday you're talking about: "my birthday", "the birthday of..", "eu nu-mi sarbatoresc ziua", "nu sarbatoresc ziua (cuiva)".