From the HP books I know that Horcruxes grant a form of immortality. So, it is not possible for someone who created a Horcrux to die from old age or succumb to diseases. I guess the same is true for living beings turned into a Horcrux. But do living Horcruxes age normally? Is it possible to cripple them or sever their limbs, head or something like that, or do they instantly "repair" themselves like inanimate Horcruxes?
Actually, I think that they are susceptible to disease, etc. Dumbledore mentioned in HP6 that using an animal that can think for itself as a horcrux is a bad idea. I doubt that he said this because it could make itself more vulnerable to attack by going out in the open. There are very few things that can destroy a Horcrux, so the reason behind this must be that a living horcrux is prone to be killed normally.
Besides, a horcrux is destroyed when the vessel is beyond magical repair. A dead snake is (probably) impossible to bring back to its original state, so the Horcrux dies the moment Nagini does. I guess crippling them is OK, as the magical world seems to be able to heal any physical injury except death. Dunno about mental injuries.
Also, Harry tells Neville to 'kill the snake' in HP7. He never mentions 'with basilisk venom/sword/Fiendfyre'. Ofcourse, Harry isn't the expert on these matters so he may have forgotten about it.
IIRC, Voldemort reinforced Nagini with his own spells, presumably to stop her from getting disease or dying early.
Of course, Neville kills Nagini with Griffindor's sword (impregnated with basilisk venom), so we'll never know the true answer.
There is nothing in canon to contraindicate a living vessel Horcrux cannot die, age, or even become ill. Remember, a Horcrux is merely a container for a portion of soul; the object in which the soul is contained (the Horcrux) is not the soul itself. It only holds the portion of the soul. So if a wizard chooses a living object as a Horcrux, the living object, the vessel, is not immune to death, injury, or destruction.
In Deathly Hallows, Neville decapitated Nagini, killing her, and destroying the Horcrux within her. Nagini did not return from the dead, nor did she repair herself after being fatally stricken by the Sword of Gryffindor.
Once destroyed, a Horcrux made from an inanimate object does not repair itself, regenerate, or otherwise overcome its destruction. It is destroyed for good.
Nagini is the only example of a living Horcrux in the Harry Potter series.
J.K. Rowling has stated that Harry Potter himself was not a proper Horcrux.
I don't think it made her immortal but I think there's a good chance she would have lived quite a long time. Nagini isn't a standard snake, she's a magic snake, she looks like a Python but has venom which no python in life has and she seems to possess an intelligence beyond that of a normal snake. I know the Basilisk could live for over a thousand years so I'm going to guess Nagini has a lifespan far beyond that of a normal human so the problem of her body dying of age isn't one that Voldemort would have to worry about. She probably didn't have many opportunities to get injured so that also probably was not a concern since she appears to be highly resistant to magical attacks, although that may actually be because she is a horcrux, I assumed it was a natural part of her magic snake biology as many magical creatures have a resistance to magic such as dragons.