"I am tired."
"is" is used when comparing two things to each other, see (tips & notes):
It is used to say that two things are equal, or to describe that one thing belongs to a class or category of other things: for example, in sentences like I am a man, the woman is a cook, he is our friend, or that is a book.
I am in deep water now, so anybody knowing better feel free to correct me:
The pronouns (mé, do, sí ...) and "tá" (or other verbs) are not really ever separated, the same applies to other forms such as "bhfuil" or "níl". "Nílimse tuirseach" or "An bhfuil tú tuirseach?".
This is related to the word order: Verb, subject etc.
I unfortunately am not skilled enough to find an answer to the word order definitely prohibiting "mé" at the end of any "tá" sentence without anything been allowed inbetween, but the basics is outlined in the two links below. (I would just kindly ask you to believe and to comply, even though that is not didactically correct. Especially the numerous samples of the second link don't support the suggestion of "Tá tuirseach mé".)
"Táim tuirseach." , "Tá mé tuirseach." and "Tá tuirse orm." would seem to be perfectly acceptable and having the same meaning. That again is an issue of languages, that you have a certain degree of freedom to say issues in different ways, no problem there.
(One place where the word order issue might be answered satisfactorily is this link, but that was way over my head:
Ó hÍde et al. (2020: 120) write in "Colloquial Irish 2":
""" English speakers have a tendency to want to say:
"Tá mé tuirseach" I am tired
Although this expression can be noted in Irish language speech, the traditional word order for the same expression would be the following:
"Tá tuirse orm" I am tired (Literally — tiredness is on me). """
Although it is worth noting that Ihde et al. (2008: 121) wrote in "Colloquial Irish" in passing:
""" Another helpful expression is "Tá mé tuirseach" (I am tired). """