What are the effects of running a PIC μC slightly above its rated working voltage?
The datasheet warns:


  Stresses above those listed under "Absolute Maximum Ratings" may cause permanent damage to the device.


But can you use a voltage above 3.6 V, but still below the absolute maximum ratings?


  This is a stress rating only and functional operation of the device at those or any other conditions above those indicated in the operation listings of this specification is not implied.


No; it's still not guaranteed to work.

But does this at least avoid the "permanent damage" mentioned above?


  Exposure to Absolute Maximum Rating conditions for extended periods may affect device reliability.


No; you still can damage your device, it will just not happen immediately.



But what will actually happen?
This is impossible to predict exactly; any part of the µC might fail, fully or partially.