The DR-680's headphone output is certainly nothing to write home about, but not to the point where you should bring an external amp. The two major flaws I've found are mediocre sound quality and a crummy potentiometer, however it is capable of driving most headphones at or near full volume. The gain range is anything but smooth, but overall it will work just fine for what you need to do at a show.
Another point to note is that I initially found the monitoring on the DR-680 to be quite confusing, probably because I didn't read the manual closely enough for quite a few outings. My mistake was that I thought the Solo function was what I needed to use to monitor specific stereo pairs, but all that function does is output a single channel to both left and right headphone channels. Instead, you need to use the front panel "Mix Level" button and lower the mix level to zero of any channel you don't want to hear. That's quite a tedious process because if you only want to hear channels 1&2, you've got to manually turn down 6 other channels (or 4 or 2 depending on how many total tracks you've got going). I much prefer the monitoring on the R-44 where a simple button press rotates between 1&2, 3&4, or all four mixed together.