Effect of time limits on problem solving
This ted talk video would probably interest you. 

It seems the current understanding is that it would depend on both the individual, and the type of task they are performing. Here are the general tendencies I've seen/been taught.

For physical tasks or tasks where the important factor is strength and effort, time constraints and rewards tend to increase the pace at which someone finishes the task. A simple example is track and field, where the contestants are generally able to perform better with the added pressure/motivation of the competition. No records are made in practice.

For more mental and creative tasks, time constraints and rewards tend to diminish one's ability to finish a task. For instance, musicians rarely make great music when they are told they have to. 

Both of the above situations however, can be affected by the person completing the task. If youlook at sporting events with very tense situations, there is an understanding that some players "rise to the occasion" and some "shrink under pressure." One's anxiety level and self confidence will affect the psychology of how they approach a task, which will certainly affect the result of that task.