If this were me, I'd make a point of "managing" the boss. When you turn your timesheet in, ask your boss to approve it right then and ask if you can fax it in yourself. Tell him you'll give him the timesheet and confirmation back as soon as it goes through. (Then keep a copy for yourself). I'm sure he'll know why you want to do this, LOL. Can't believe he would say no.
If he does say no and then forgets again, consider filing a written complaint with your HR department or even your state's Attorney General. Of course, you'll probably want to work elsewhere if you have to do that - can't imagine working conditions would be very pleasant after filing that kind of complaint. It won't be happily received by the Company's poweres that be - trust me!
Anyway, try to take control of the problem if you can (by faxing it in yourself). IMHO, this would be better than having to involve others. I'd also suggest the same thing to the co-worker who has also had the problem.
Best wishes -