What are the consequences of running a component above the recommended operating conditions but below the absolute maximums?
Absolute maximum is the minimum of what might be expected to damage a device. 

Recommended operation conditions is when the device will work as specified.

In between, the device might not be damaged but it might not work as well as you expect it to (or might not work the way you expect it to which is considerably worse).

In your scenario example, excessive voltage output droop is to be expected, and probably other things like linearity, etc. Not to mention heat.