Shinigami means 'God of Death'. I think it is reasonable to assume that God is above death or birth.
However to look at this objectively in the light of the manga, the shinigami is summoned by the Dead Demon Consuming Seal. Since it only acts as a container of souls, any impact on physical world should not reflect permanently in the spirit world. Additionally, it might be possible that on each invocation of the seal, a different shinigami is summoned. In which case, we cannot really say anything about the fate of a particular shinigami.
I will also propose another hypothesis. I think every person has their own personal manifestation of the shinigami in the Naruto universe. This solves a few problems. Two possible ways to go about it:
Every user summons a different shinigami. The shinigami devours the user's soul as well. Since a body cannot last long without its soul, the body will die soon. This raises two possibilities:
- The shinigami dies along with the user. So every shiniami is different. There are as many shinigamis as there are people in the physical world.
- The shinigami is immortal. There may be one or more of them.
Both possibilities ensure that any other person's summoning is not affected.
The user can never really summon the real shinigami. This means that every user gets a clone from the spirit world. But this could mean that any impact on a clone might reflect on the real one. Again we must assume that all shinigami must be immortal.
There exists just one real shinigami. And every user summons it with the seal or mask (more on this later). And it must be immortal because of reason stated in point 1.
The third idea seems best supported. The fact that there are several Oni-masks in the Uzumaki Clan's Mask Storage Temple seems to suggest that they might be representations of Gods. Thus the shinigami mask would then be associated with the 'God of Death'. Then it would mean only one 'real shinigami'. Every user summons it with either the seal or the mask. And it is immortal.