Judging from the Feign Death spell in Tome and Blood, there is at least resistance to energy drain, and 1/2 damage from ability drain or damage. Physical damage is also halved. The Magic Jar trance should give at least this much protection.
Another question: if the body I am possessing takes ability damage, does it stay with the body, or with me?
I'm tempted to say that physical damage would stay with the body, and mental damage would stay with the possessing spirit. And that would suggest that the Magic Jar trance should give protection from mental drains and ability damage.
So a good tactic for a Master of Shrouds character who has summoned too many Allips... cast Magic Jar and wait out the problem.
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Feign Death
Necromancy
Level: Sor/Wiz 3
Components: V,S
Casting Time: 1 action
Range: Touch
Target: One willing, living creature whose level or Hit Dice do not exceed the caster's
Duration: 1 hour/level (D)
Saving Throw: None
Spell Resistance: Yes
You put the recipient into a deathlike condition. At your option, this can be a cataleptic state that is impossible to distinguish from death, or the subject can appear to be in a coma or deep sleep. Although able to smell, hear, and know what is going on, the subject is blind and has no sense of touch or pain. The subject does not need to eat, drink, or breathe.
While the spell is in effect, the subject is immune to subdual damage, and any normal or ability damage inflicted is reduced by half. In addition, the subject is immune to mind-affecting attacks, paralysis, poison, disease, and energy drain. The effects of any poison or disease already affecting the subject when the spell takes effect are halted until the spell ends. if the subject has any negative levels when the spell takes effect, the saving throw to remove it is delayed until the spell ends.