Should you let your opponent know if they have forgotten to press their clock? I'm in two minds here. 

The good sportsman says you should always let your opponent know (maybe limited to once per game). 

The more hard-nosed part of me says that managing the clock is part of the game and rules like anything else. If you are playing seriously then you should treat the clock no differently to for example, the 'touch-move' rule and expect no help from your opponent.

Does it matter whether one is playing a club game or a tournament with prize money at stake?
In my personal opinion, if you have noticed that your opponent has forgotten to hit their clock, the unwritten rules of good sportsmanship dictate that you should advise your opponent of such. It can become distracting to both players, though, if you have to do this more than twice.

So the real question comes down to the legal technicalities. 

Are you even allowed to make your move before your opponent hits his/her clock? The USCF rules, at least, are not very clear about this. But it does not seem to be illegal at all. Many people feel it is bad form to move before your opponent has pushed the clock, but such a rule would be unenforceable from a practical perspective. 

What about in sudden death? Tournament Directors and Arbiters are generally directed to not interrupt the game to point out illegal moves when in sudden death. So even if this were a rule, it likely would be set aside in sudden death.

Other than information about time delays and time controls here is what the USCF says about the clock:

6.) Except for pressing the clock, neither player should touch the clock except:
   6a.) To straighten it.   
   6b.) If a player knocks over the clock a penalty may be assessed.   
   6c.) If your opponent’s clock does not tick you may press his side down and
        re-press your side; however, if this procedure is unsatisfactory, please
        call for a director.   
   6d.) Each player must always be allowed to press the clock after their move is made.   
   6e.) A player should not keep a hand on or hover over the clock.


Back in 2008 there was a bit of a controversy about these questions regarding the clock. For the most part things have just been left up in the air. So in summary, it is not your responsibility to advise your opponent, many people feel it is good form to do so. But if it appears that your opponent is habitually forgetting to press the click, I would just continue on with the game and not advise him or her more than twice. There is no reason that this should interfere with your enjoyment of the game and ability to concentrate. Ultimately, in the USCF, the rules put the onus on the individual to press his own clock.

Kasparov v. Karpov, 1987 (pointed out by Tony Ennis) from the AP Archives


  For almost three minutes, members of Kasparov's delegation looked on
  helplessly as his time slowly ticked away. Under the rules, neither
  they nor the match arbiter could warn Kasparov of his mistake. When he
  finally noticed, Kasparov had less than a minute left for 14 moves. As
  he desperately played move after move, Karpov closed in with a
  crushing mating attack. "It's really a very bad psychological shock,"
  Spanish chess writer Fernando Urias said. "Kasparov has not only lost
  right at the start of the championship, but he also lost playing with
  the advantage of white."