I'm not sure if the output level from the bass is enough to make the headphones work. Another issue is that most headphones are stereo, and the output jack on a bass is mono, so it will only come through one side.
There also are headphones with built in amp and aux input, so you can mix the bass and the other source.
Both amp and aux have their own volume pot on these, and there's a mono/stereo switch that doubles as an on/off for the amp. This way it can also be used as a regular headphone.
I've used it extensively on a bass with passive pickups, and with stronger signals from other sources. Works a treat.
Mine's without brand name, so I can't point directly to the right product, but a search should help.
somebody mentioned a headset with built in amp.. ping @adouglas??
ive used both the korg amplug and a korg pandora box with good success.
as for the headphones directly into the amp, i think you might experience difficulties there. the output amp in the bass isnt meant to drive so low impedances and might get damaged from trying to supply the currents needed for getting what you want.
also, if im not wrong, the input jack uses the sleeve to shortcircuit a switch that turns the preamp on. so when you plug in a jack with tip, ring and sleeve the amp might not even turn on.