Question: Is tumira only for where you permanently live?

No, “tumirá” does not mean where one lives “permanently”, but only “for a certain or an extended period”, or possibly, “periods”. “Tumirá” is the verb “to live/reside” and it is also its past tense form. (Tumirá: tumirá, tumítirá, títirá) TUMIRÁ kamí sa Virginia. = We LIVED in Virginia – This means that “there was a time or there were times when we LIVED in Virginia”.Noóng pumuntá kamí sa Pilipinas para MÁGBAKASYÓN/MAG-VACATION, TUMIRÁ kamí sa hotel NA malapit sa Mall of Asia. = When we went to the Philippines for vacation, we LIVED in a hotel that is near the Mall of Asia.

It is “nakatirá” that could be understood as one’s permanent residence. “Nakatirá” is a state/attribute that can function as a pseudo-verb. NAKATIRÁ kamí sa Virginia. = We LIVE/ARE LIVING in Virginia.TUMIRÁ kamí sa Virginia noong/nung nagtatrabaho PA akó sa Pentagon. = We LIVED/RESIDED in Virginia when I was STILL working at the Pentagon.NAKATIRÁ kamí sa Virginia noong/nung nagtatrabaho PA akó sa Pentagon. = We WERE LIVING/RESIDING in Virginia when I was STILL working at the Pentagon.TUMIRÁ akó sa Virginia noong/nung nagtatrabaho akó sa Pentagon, pero ang pamilya ko ay sa New York NAKATIRÁ/TUMIRÁ. = I LIVED in Virginia when I was working at the Pentagon, but my family WAS LIVING/LIVED in New York.NAKATIRÁ ako ngayón sa Virginia habang nagtatrabaho sa Pentagon, pero ang pamilya ko ay sa New York NAKATIRÁ. = I LIVE/AM LIVING now in Virginia while working at the Pentagon, but my family LIVES/IS LIVING in New York.We don’t have an exact Tagalog word for “permanent”. We use the Spanish "permanente” for that. “Palagi” is the Tagalog word for “always”, but since it is based on frequencies, it is at times used to mean “almost always/very often”, too.Permanent residence = permanenteng tírahan.We also express permanence with additional information.Sa Virginia NA ako TUMIRÁ simulâ/mulâ noong/nung ipinanganák ako. = (Lit. It is ALREADY in Virginia where I LIVED since birth) I HAD BEEN LIVING in Virginia since birth. Sa Virginia NA ako NAKATIRÁ simulâ/mulâ noong/nung ipinanganák ako. = (Lit. It is ALREADY in Virginia where I AM LIVING since birth) I HAVE BEEN LIVING in Virginia since birth.Sa Virginia NA ang TÍRAHAN ko simulâ/mulâ noong/nung ipinanganák ako. = (Lit. Virginia is ALREADY my PLACE OF RESIDENCE since birth) Virginia HAS BEEN my PLACE OF RESIDENCE since birth.

thanks!

Juantutri is correct.
You can also use "tumuloy" or "nanuluyan" which can mean "stayed" (tumuloy can also mean proceeded while nanuluyan is just stayed).Nakatira kami sa Viriginia./Taga-Virginia kami. Noong pumunta kami sa Pilipinas para magbakasyon, nanuluyan kami sa hotel malapit sa Mall of Asia.
We live in Virginia./We're from Virginia. When we went to the Philippines for vacation, we stayed in a hotel near Mall of Asia.

What are some situations where one of the words (tumuloy, nanuluyan, tumira, nakatira) would be used rather than the others?

Both “tumulóy” (tumulóy, tumútulóy, tútulóy) and “manuluyan” (nanuluyan, nanúnuluyan, manúnuluyan) have something to do with temporary accommodation. “Tumuloy” refers to going to the place of interim or temporary accommodation from one’s most recent point of origin, to include transport terminals. Sa hotel kami tútulóy galing sa airport. = We are proceeding to/checking-in at a/the hotel from the airport.
Tumulóy muna kayó dito bago kayó umuwî. = Visit us/with us here first before going home.
Sa bahay ng lola ko akó tumútulóy tuwíng pumúpuntá akó sa Manila. = I proceed to/stay at my grandma’s house/place every time I go to Manila.“Manuluyan” refers to the period of stay at the temporary accommodation.Dito akó sa hotel na malapit sa bus terminal nanúnuluyan (sa) ngayon``. = Here at the hotel near the bus stations is where I am staying (as of) now.
Pwede ba akóng manuluyan diyán sa inyó ng mga dalawáng araw? = May I stay there at your place for a couple of days?Unless the number of days of stay is mentioned, “tumuloy” is normally understood to be for a short time and “manuluyan”, for a relatively longer time. When using “tumirá” or “nakatirá”, normally, you are implying that you consider that place as your official residence at that time. However, whether you use a “tulóy” or a “tirá” word, what you actually mean could be understood based on context. For example, if someone knows that you are at a certain place only for a week, and you are staying at a hotel, whether you say “nakatirá ako sa hotel” or the more appropriate, “nanúnuluyan ako sa hotel” won’t really matter as either way would be understood as a temporary stay.