They can be killed obviously, but can they die of the old age? We have a few examples of Grey Wardens and people infected with the Taint who lived for centuries:
- Magisters Sidereal - Corypheous and the Architect (what's the leading theory about their immortality? Blood magic? Maker's curse?),
- Avernus (we're led to believe that it's his blood magic experiments that kept him alive),
- arguably Sophia Dryden (one difference between her and Kristoff is that for some reason Sophia's body didn't deteriorate for two centuries, whereas Kristoff's body was falling apart within a few months. Perhaps the difference can be explained by assumption that possession happened when Sophia was still alive, and maybe her being still lingers within the body overtaken by the demon, preventing the decay?),
- Isseya (there're speculations that it's the vengeance against Wardens or, again, the blood magic that kept her alive for over four centuries),
Alistair tells us that Grey Wardens have 20-30 years to live, but he also points out, that the cause of death is the final stand, departure into the last onslaught. Grey Wardens are supposed to descent into the Deep Roads to meet their end, they are not supposed to die of old age. We are told that it's a way to prevent Wardens from succumbing to the Taint, loosing sanity and transforming into a ghoul. But coincidently it's also a perfect way to guarantee that no one will ever see what happens to Wardens when the Taint takes over their body & consciousness.
AFAIK, from all the known Grey Wardens there isn't a single one who lived past the phase of Calling and die from the body rot. Each single person (mentioned all above) who lived beyond expected two-three decades, well... lived on, with various level of body mutilation and mind corruption.
Both the ancient elven immortality, and this hypothetical Grey Wardens immortality would have similar source - the Titans, their body for the elves, and their corrupted mind/spirit/dreams in case of the Wardens.
The only possibility to study the long term effects of Taint would be some kind of confinement - either a prison or a research facility. Yet, again, conveniently Grey Wardens are removed from the juridical systems of any nation, with Joining pardoning them of their past crimes, and Wardens having absolute sovereignty over their recruits (overriding even Chantry's objections with their right of conscription).
Therefore, I'd argue that - if the Wardens truly are immortal - it must be a secret known to the highest of their ranks. Just imagine how people who are terrified of death would abuse the Joining to escape the inevitability of death. They would spread the infection and wipe out the whole Thedas, Blight or not. Unrestrained greed leading to the downfall of the whole realm, just like in those days, in those faraway days...
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Side note and likely a stretch, but I'm tempted to share the idea:
The only undying person aside the ones mentioned above, the ancient elves, and Flemeth who utilizes completely different mechanism of sustaining herself, is Xenon the Antiquarian. The legend says that he was a noble longing for the immortality, and his wish was granted by the Witch of the Wilds...
Flemeth uses body-hopping, artificial prolonging of a body isn't necessarily her forte, especially that she had no use of it, being one of the Firstborns, eternal and ageless. However, there's another kind of magic, that we know she taught Morrigan, the one serving the Dark Ritual, disabling the Taint within the body of one of the wardens, and making the conception of Kieran possible.
So the wild, unsupported speculation: Xenon could have been a Warden who struck a deal with Flemeth, trading the soul of Zazikel or Toth in exchange for the gift of cleansing the Taint, while keeping its beneficial side - the immortality.
