The One Ring extended Gollum/Sméagol's life and vitality by 478 years and Bilbo's vitality by 60 years. Did it have the power to make them truly immortal if they had kept it, or did it just grossly slow down the aging process? The ring was made by a Maia for a Maia (spirit), not a mortal.
A mortal wearing a ring of power does not age, nor can he die of natural means (Gandalf says this in FOTR when he tells Frodo of the rings of power)
For evidence see the Nazgul, who were men that lived in the middle centuries of the 2nd age, they survived until the end of the 3rd age after being gifted rings of power, a period of around 4-5 thousand years.
Also see Gollum who lived 500 years after taking the ring.
However, whether the rings were capable of immortality without corruption is debatable, I find it hard to believe that the rings were ever created with the intention of corrupting men to evil (Sauron was involved in their creation (except the 3 elven rings)) but I think that it was the creation of the One that corrupted the mortals wearing the rings.
That said the dwarves were originally made by Aule to be resistant to corruption and the devices of Melkor, and as such they resisted the rings more than men, and did not recieve immortality from them. However the Dwarven rings corrupted them with their specific weakness (Gold lust).
Finally any mortal wearing a ring could still be killed, and regardless of how long they lived and their power they would still go to their respective fate. Men (hobbits as well) would go to "Mans Doom" and leave Arda, and Dwarves would be gathered by Mandos.
The fact that a man with a ring of power can be killed is evidenced in several places:
- Isildur's death at the Gladden Fields
- Deagol's death in the same place
- Gollum's death in Mt.Doom
- Angmar's death at the battle of the Pelennor Fields
So in answer to your question:
- Men with a ring of power do not age
- Men with a ring of power can be killed
- Dwarves with a ring of power age normally
- Maia and Elves are of course immortal anyway and therefore Rings have no effect in that regard.
Immortality in Tolkien is very different to not dying.
To be immortal means that your spirit is bound to the world, the Music of the Ainur is as fate to it, and it does not leave the world until the end of the world, beyond which its not known what happens to it.
The doom of the Elves is to be immortal, to love the beauty of the world, to bring it to full flower with their gifts of delicacy and perfection, to last while it lasts, never leaving it even when 'slain', but returning – and yet, when the Followers come, to teach them, and make way for them, to 'fade' as the Followers grow and absorb the life from which both proceed. (Letter 131)
To be mortal means that your spirit is not bound to the world, it can shape it's own destiny beyond the Music, but it must die and leave the world after a short time; beyond the end of the world it is destined to take part in the Second Music of the Ainur.
Therefore he willed that the hearts of Men should seek beyond the world and should find no rest therein; but they should have a virtue to shape their life, amid the powers and chances of the world, beyond the Music of the Ainur, which is as fate to all things else ... It is one with this gift of freedom that the children of Men dwell only a short space in the world alive, and are not bound to it, and depart soon whither the Elves know not ... Yet of old the Valar declared to the Elves in Valinor that Men shall join in the Second Music of the Ainur; whereas Ilúvatar has not revealed what he purposes for the Elves after the Worlds end ... (Silmarillion, Of the Beginning of Days)
Tolkien is quite clear in several places that only Eru can change a being from mortal to immortal or vice-versa, for example:
Immortality and Mortality being the special gifts of God to the Eruhini (in whose conception and creation the Valar had no part at all) it must be assumed that no alteration of their fundamental kind could be effected by the Valar even in one case: the cases of Lúthien (and Túor) and the position of their descendants was a direct act of God. (Letter 153)
Therefore the One Ring is quite incapable of making a mortal creature become immortal.
This can be seen in Gandalf's words to Frodo concerning the effects of the Rings on mortals in Shadow of the Past (my emphasis):
A mortal, Frodo, who keeps one of the Great Rings, does not die, but he does not grow or obtain more life, he merely continues, until at last every minute is a weariness.
And is completely consistent with statements elsewhere, for example the following footnote to Letter 131:
The view is taken (as clearly reappears later in the case of the Hobbits that have the Ring for a while) that each 'Kind' has a natural span, integral to its biological and spiritual nature. This cannot really be increased qualitatively or quantitatively; so that prolongation in time is like stretching a wire out ever tauter, or 'spreading butter ever thinner' – it becomes an intolerable torment.
A mortal who uses the Ring is therefore still a mortal: merely one who's lifespan has been prolonged (indefinitely and intolerably), but they are still subject to the fate of mortals, which remains completely unchanged.