Can a greasy chain cause the chain to come off the rings? I heard once that you should clean your chain with a light degreaser whenever it looks especially dirty or at the very least once every few months. But that doesn't account for frequency of rides or miles ridden! 

I've been riding my road bike over 100 miles per week for the last month and have been meaning to clean my chain anyway... and then...

The past week my chain kept popping off the chainrings. Four times out of five it was while shifting from the small ring to the big one. I believe it also happened when accelerating from a pretty low speed, like just at the crest of a very steep hill.

I'm going to clean my chain anyway, but I'm just curious: could a very dirty chain be causing it to pop off the rings? Or could this be a problem of technique? Or do I just need a general tuneup?
No.  A greasy chain won't cause the chain to pop off, and it would have to be exceptionally dirty for dirt to cause this.  (I've ridden many miles with greasy, dirty chains.)  Generally a chain comes off due to maladjustment or because of a worn component (usually a cog).

Poor technique can contribute to the problem, but the technique either needs to be pretty bad or there needs to be another contributing factor.