Should I remove and reinstall a Jacuzzi / whirlpool tub when replacing the paneling around it with tile?
As I mentioned in a comment, this is largely a matter of personal preference, available resources, etc. What I would want to do is remove the tub. I can't say if that's what I'd end up doing. My reasoning:

Deck tile should really extend underneath the tub flange. It won't look great if you have to fit tile around it and grout to it, even if you caulk. A clean overlapping edge looks best.
Therefore, the backer should also extend underneath the tub flange, and the solid wood decking should be removed.
As you stated, the rather old plumbing should be upgraded. This is best done with the tub out and the connections and space more accessible.
The tub is less prone to damage during tiling if it's out of the way. You could certainly protect it with cardboard and blankets, but even then scratches and chips are possible.

Note that the tub doesn't actually rest on its flange. It's not designed to carry weight. You'll need to plan well and possibly shim up the tub base to meet the height of your tile.