Are heavily worn rims safe to use on trainer? I've got a wheelset with heavily worn rims.  This was a cross wheelset used for riding in pretty sloppy conditions all the time.  This was a rim brake based wheelset and used with cantilever brakes.  Now the braking surface is rather concave. It's worn enough that I feel like riding on the road would be unsafe due to risk of failure.

I was considering just using the wheels on my trainer, where I would no longer need to do much in terms of braking (and won't actually be going anywhere).  Do you think this is safe, or should I just go out and buy a cheap / heavy wheelset for indoor training?
I don't see why not, unless you plan on braking while on the trainer? Theoretically you could still have a rim failure because the wheel/tire pushes against the resistance wheel. However, this is a constant tension, not sudden impacts. If a spoke were to become loose though, the rim could fail and the wheel could implode, but since your on a trainer it's not really all that dangerous (besides picking bits of wheel out of your drivetrain).