If the UR22 will drive the HD600, and it looks like it will according to clouden, then no, you don't need a dedicated DAC.
The conversion as carried out by any halfway competent audio interface will sound at least as good. If not better.
How can I be so sure?
One of my interfaces is a 4 year old MOTU Ultralite. It offers 10 channels in and 14 channels out. So when I first got it I looped the audio up to 5 times and recorded the result. That's ADC>DAC>ADC>DAC>ADC>DAC>ADC>DAC. I'm not going to claim no one can tell the difference between the original and iteration 5 (to avoid a fight) but it must be extremely difficult. I couldn't. Not by ear alone.
Fortunately some kind soul on gearslutz did the same and posted his test files. So you can also test yourself.