What happens if a player suffers a time penalty but doesn't have enough time left? Wikipedia's article on threefold repetitions gives a story about an incorrect threefold claim by Karpov against Miles:


  The clause about the right to capture en passant and the right to castle is a subtle but important one. In a game between grandmasters Anatoly Karpov and Tony Miles (Tilburg 1986), Karpov had less than five minutes remaining on his clock in which to finish a specified number of moves or forfeit the game. He claimed a draw by repetition after checking his scoresheet carefully, whereupon it was pointed out to him that in the first occurrence of position, Black's king had had the right to castle, whereas in the second and third it had not. Tournament rules stipulated that a player be penalized with three minutes of their time for incorrect claims, which left Karpov's flag on the verge of falling. By then, Miles had taken the draw ...


What if, instead of five minutes left, Karpov only had two minutes? Does he instantly lose on making the incorrect claim?
According to the FIDE Laws of Chess:


  12.9 Options available to the arbiter concerning penalties:
  
  12.9.1 warning,
  
  12.9.2 increasing the remaining time of the opponent,
  
  12.9.3 reducing the remaining time of the offending player,
  
  12.9.4 increasing the points scored in the game by the opponent to the maximum available for that game,
  
  12.9.5 reducing the points scored in the game by the offending person,
  
  12.9.6 declaring the game to be lost by the offending player (the arbiter shall also decide the opponent’s score),
  
  12.9.7 a fine announced in advance,
  
  12.9.8 exclusion from one or more rounds,
  
  12.9.9 expulsion from the competition.


In such situations, for a first offence, the arbiter would always add time for the opponent. He would never reduce the time for the player. Only if it were a third or later infraction of the same rule and the player had been warned would more serious action be taken like defaulting the player.

There is a general principle in the arbiting world, which is seen in the move away from allowing quickplay finishes, that the actions or decisions of the arbiter should not decide the game. Reducing the player's time to less than zero would do just that and would never be done.