If a player must lose Grit for each card they cannot draw when their deck is empty and their play area is exhausted, the rulebook states:

• "If your entire deck is in your play area, lose one [Grit] for each card that you cannot draw."
  (page 13)

There is no rule in the provided context that prevents a player's Grit from going below zero as a result of this penalty, nor is there any mention of an additional penalty for having negative Grit in the standard (non-cooperative) game.

Summary

• Yes, this penalty can cause a player's Grit to go below zero.
• There is no additional penalty specified in the provided rulebook context for having negative Grit in the standard game.

Source: page 13

If you are playing the cooperative mode, the team loses if Grit drops below zero (page 27), but for individual players in the standard game, the only effect is a lower (possibly negative) Grit score.

Source: Core Rulebook (pp. 13, 17, 24, 27, 3)