As far as I know, Yes, animals will age and die as players do however we don't yet know of the rate of aging. I would assume that different breeds or types may age at different rates and it might be possible for a breed of animal to last more than one Mann lifetime but I would think that would be rare, but no detailed information has been released on animal aging yet (ie. Pets vs Mounts vs Burden animals aging rates).
It is also possible and very likely that they can be killed whether it is out of spite or during combat, so animal breeding and taming will be very useful.
This mechanic (along with others) is one reason that many say the tiers are not pay to play. Anything you get from them, be it capes, mounts, or even titles can be taken away or lost, though the latter one will take a little more work.