When somone is there at Yitzias Neshomah (time of death) there is a requirement to rip Kriyah what happens if this situation occurs on Shabbos? Do you rip Kriyah or do the laws of Shabbos prevent you from doing so?
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As far as I can tell, definitely not - both on Shabbos and Yom Tov.
The Mishna and Gemara in Shabbos are discussing whether the one did it is exempt from bringing a Korbon (after the fact). The Gemara ends up saying that it depends on whether there's an obligation to tear for that relative. If there's an obligation, then the one who tore is חייב, if not the one who tore is פטור.
However, one should definitely not do it Lechatchilah. (confirmed source: here page 9)
Interesting aside: you specifically asked about death. But the same would also apply to tearing for the Kosel. Which is one of the ways to avoid having to tear for the Kosel (i.e. by having the first visit done on Shabbos or Friday Afternoon) [see here for source].
Two things I'm not sure of though, which I would hope someone could cover in the comments: