The really bad problem would be if you had a constant DC level on the headphones, which will sound pretty much the same, because there's still that sharp vertical, but this time no decay back to 0. Whether your headphones can stand a constant DC level depends on the level and the headphones themselves, but the worst part is that you end up distorting the music coming through, which can be bad for the headphones in their own way.
So yes, do a DC measurement on the amp. If it's over about 20 mV, something's wrong. In fact, with a low-offset chip like the 8610, if it's over about 5 mV, be worried. All of my 8610/20 amps have been in the 1 mV area for DC offset, IIRC. It's a wonderful little chip that way.
It's not that 5 mV is going to hurt anything, but rather that it's abnormal for this chip, so it's worth seeing if there is a real problem or if you've just got an out-of-the-ordinary situation somehow.