Would you recommend me remove the dead foliage, or leave the foliage on?

Dead foliage will not help with winter protection. You're protecting it anyway, although I don't know if I'd use burlap. Prune away, just don't prune anything green.

Will it harm cold hardiness? Or is it neutral in a sense?

You can leave dead foliage on, but I'm not sure why you would want to...Your palm will look nicer if dead fronds are pruned off. As mentioned above, it won't help with winter protection. I've heard that leaving dead fronds on a washy helps to insulate them once they get that skirt of dead fronds because it protects the trunk. However, if you're protecting it anyway, then you don't need a "skirt" of dead fronds on your needle.

I agree, dead fronds can definitely be cut off. It won't do anytime to make it less cold tolerant. Never cut off fronds that are green and healthy though since your palm can use as many of those as possible to build up strength to handle the winter cold or any other possible problems along the way.
You can definitely make an argument about keeping washy fronds because of the skit it makes like islandbreeze says, but needles never develop that skit so leaving dead fronds on it will just make it look bad and will not insulate it in any way.
Good luck!
-Alex

Use frost cloth in place of burlap, it breathes and helps shed snow and freezing rain, thus keeping the crown dry.