Why do higher incentives lead to lower performance for non-rudimentary tasks? I have watched a number of times this excellent video where Dan Pink discusses the science of motivation. The video states that the higher incentives, the lower performance for non-rudimentary (not easy and quick) tasks. 


Why is this true for non-rudimentary tasks, but not for rudimentary tasks? 
Does anyone know any theory, which can help explain this?
I think you have to be careful with the proposition that "rewards do not increase performance on non-rudimentary tasks". The experiments that Dan Pink cites involve experiments where participants are in a room and are supervised by an experimenter while they complete a task. This social pressure by the experimenter may well be enough for participants to be focused and diligent in completing the task. 

But in the real world:


people often have time to prepare for situations that involve high stakes rewards. For example, performance on exams often have major rewards tied to their outcome. A major goal of exams is to motivate students to study hard for the subject matter.
effort and time on task is often not fixed. Employees can do overtime if the task is hard but the rewards are good (or if the pay implies that the extra effort is expected). They can engage in less off-task behaviour. 
people choose the tasks they do at all. When people choose a career, it is assumed that among many other things, that many will consider the probability of getting a job and the likely salary and conditions (i.e., rewards). When a person chooses to work for a particular employer, salary relative to other potential employers is often an important factor. Thus, rewards can influence whether an individual engages in a particular activity at all.


Thus, I think one needs to be careful when generalising the meaning of these experiments to work and other contexts.

In terms of theories that might explain the results:


The Yerkes-Dodson curve and related models suggest that the relationship between arousal and performance is an inverted-u relationship and that the peak of the inverted u varies with task difficulty (i.e., greater arousal is optimal for simpler tasks).