Not all pins are even active at any given time - some are reserved for features that your CPU may not have, so if those kinds of pins are dirty or even destroyed, nothing will happen.
Generally it's not a good thing though. I don't think they even make conductive thermal paste anymore so it shouldn't short, but it can cause bad contact. Either due to to the paste's insulation properties or because the paste is pretty thick and can cause some damage to the ultra-delicate motherboard pins.
So maybe nothing would happen, and maybe the computer won't boot, or maybe it will become unstable after a while or under heavy load...
It's unpredictable. If you are unable to clean the paste, then you can only hope for the best I suppose.
As for the buzzing sound, no clue, that could be anything but I doubt the CPU would behave oddly in such a specific way.