I don't think they should, but Quin Snyder could really be on the hot seat if the Jazz don't win this series.
He's been a great coach, but they've also had some relatively poor playoff showings and haven't made the WCF in the last 5-6 years. Lots of 1st and 2nd round exits. Even last season, they lost to the Clippers even without Kawhi for the final few games. Now if they lose this series despite Doncic missing the first 3 games of it, I think they'll consider a change.
There are other issues on the Jazz, but we have seen coaches get moved or transitioned (Dwayne Casey, Brad Stevens) despite having relative success especially in regular seasons.
I think it's a foregone conclusion though Nash is gone. I mean, I guess if BKN wins say the next 2 games and shows some fight, maybe they reconsider. But unless they pull off the 3-0 comeback I don't see Nash returning. I think deep down even Nash knows this too.