Hi Epitectus,
Strange but intriguing question indeed. I have no experience whatsoever in string theory (apart from some very introductive studies in supersymmetry), so I cannot comment on how time is treated there.
In the other large and common physical theories, time and mass are totally uncorrelated(*). Time is merely an parameter used to describe the evolution of a system.
Sidenote:
(*) okay, there is a so-called "energy-time" uncertainty relation in QM, but there are various interpretations of this relation and my personal interpretation is that the time that appears there is not the same concept of time as used elsewhere in QM.
Now, this is what "the theories" state, and this doesn't necessarily mean that it is correct

.
There still is IMHO a problem in describing time in physics; as einsteinsdream noted, space and time are thightly connected. Personally I wouldn't stress that as much as people do when studying the theory of relativity (in which time is indeed considered as an extra dimension with a strong interaction with the spatial dimensions). The problems arise in the study of elementary particles, where nature appears to have a "time arrow": time always evolves forward, it can never evolve backwards. This means that time cannot be treated plainly as an extra dimension, since you're not free to move around in this dimension the way you like : you can go anywhere in space, but only forward in time.
Concerning the mass of time: the first problem you have is to define what time exactly is. Personally, I'd describe the evolution of time as seeing the evolution of systems. Since in the process of "seeing" things, light and hence photons are involved, you could perhaps somehow couple time with fotons (which have a restmass of zero).
This picture is far from complete (and note that it is my personal interpretation). But no I come to think of it, why would you want time to have a mass anyway ? This would surely introduce more problems than answers I think

.
Bye!
Crisp
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"The best thing you can become in life is yourself" -- M. Eyskens.