When I attempt to listen to music or play videos while my N800 is charging, there is a background buzzing or rapid "clicking" in the headphones. It only persists while the unit is charging.

I did a search and didn't find any other comments about similar problems. Has anyone else come across this problem? If so any solutions? My firmware is up-to-date.

Maybe the headphone ground output is not connected to the GND signal coming from the charger. You can concievably check this by using a continuity meter to test whether the two grounds are connected.
This is pure speculation on my part though, and stems from my _very_ little experience in electronics.

Disclaimer: Don't blame me if you try this and your N800 turns into a pink marshmallow--or if it stops functioning correctly.

OH MY GOD MY N800 MELTED! Thanks, Convulted

The buzzing is not continuous...it comes and goes. The intermittent aspect has nothing to do with my using or touching the unit. It's just sitting on my desk untouched. I wonder if it has to do with the battery level?

Maybe the charger is playing up. I've never experienced it myself though under normal usage (I also have a 770 not a N800, but that shouldn't matter too much).
However, I do experience a similar buzzing sound when I connect the 770 to my car stereo and ground it incorrectly, hence my suggestion.

* Edit *

The buzzing I experience only happens when there is sound output.