Are there any disadvantages with working for non-tenure track professors? Is it possible that they get fired and you need to restart with someone else? Do they have less funding? Any other drawbacks?
Currently applying to phd programs.
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This answer might only apply in the US and only applies in certain very restricted situations. I know a few people (maybe half a dozen) in CS who would be suitable as advisors, depending on your interest.
There is a special category of faculty at some top US universities called, perhaps, "Professor of the Practice". They aren't tenured (or tenurable), but serve on long term renewable contracts (about seven years). The ones I'm thinking of hold solid doctorates themselves, but are primarily responsible for teaching undergraduates. They all do research themselves, but it is more likely to be in something like pedagogy of their field. They are quite prominent in the (CS) profession and are frequently seen at conferences, and such. For these few people, having them as an advisor would be safe enough.
I don't know if there are similar situations elsewhere and, even here, they are a bit uncommon, though they tend to occur at top universities (Stanford, Duke, ...).
And note, that I haven't asked any of them if they have done such a thing.
But, for the general case, it would be quite risky. They might leave for other reasons than getting fired. If their main subfield isn't mainstream you might have trouble finding another advisor to take their place, necessitating a change of field or a move. Much better to have someone with stable employment and the time necessary to give you proper guidance. Even non-tenured, but tenure track, people can fail on that.
The one concern I would have with pursuing a doctorate with a non-tenure track professor is the potential they could leave the university before you finish. Then you would have to find another faculty member to advise your research. Tenured professors do leave universities, but the likelihood is less with more secure employment and a commitment by the university.
There are always risks with any advisor. For instance, I had one die on me.
But it's perfectly possible to have advisors who aren't tenure-track, especially if they are primarily researchers. For instance, I did my MS with a lab director who was also a practicing MD, and taught only an occasional course. He had several PhD candidates working under him.
Another who I seriously considered doing a PhD with was primarily a researcher at (large tech company), who likewise taught the occasional course in his field.
You are getting politically correct but misleading answers. The canonical answer to your titular question is usually just: no.
One way to think about it: would you perhaps like to be tenure track faculty at a research university some day? If so, make sure to choose as your advisor someone whose research is top notch and well known. With very rare exceptions, these people are all tenured or tenure-track faculty.