This simply isn’t defined anywhere in the rules. There is no technical definition of “immunity,” and it is very likely that the original authors never even considered the question, since in Core, it would never make a difference. (And, truly, even outside Core, it is very, very rare for it to matter anyway.)
Trying to read between the lines in the rules for poisons, saving throws, and so on, is a mistake: none of those address immunity explicitly, and we know immunity is making some kind of exception to those rules. Where the exception comes in and what it applies to makes all the difference to this question, and the rules have zero guidance for us on that.
The one thing we do have is neutralize poison (with thanks to Peregrin Took for pointing it out), which says
The creature is immune to any poison [and] the creature need not make any saves against poison effects applied to it during the length of the spell.
(SRD → Spells → Neutralize Poison)
To me (and not to Peregrin), “need not” is quite distinct from “does not” or “cannot,” and directly implies that making the saving throw is an option. However, this rule only applies to immunity from the neutralize poison spell, because that’s where the rule is found. It’s a specific case that could be an exception—maybe every other type of immunity works differently and neutralize poison is just offering an extra, special benefit of also having the option of skipping the saving throw (or having the effect of always skipping the saving throw, whether you class that as a “benefit” or not). Maybe. Or otherwise, perhaps, it is an example of what was intended for immunity in general, and we should apply the same rule everywhere. Unfortunately, this is still very unclear, and only helps us somewhat.
So you are simply going to have ask your DM, or if you are the DM, you are simply going to have to make a ruling.