Anyone else also notice that Harvey Dent is still clean shaven after he gets burned?

Blame the comics writers.

I would think he shaves his "good" side and that the hair follicles were destroyed on the "bad" side. He's impeccably dressed and tidy on the "good" side, too.

How long was he like that?

I got the impression it was only a day or two

No way his exposed eyeball would continue to work for much longer than that

He wasn't alive long enough to have any kind of sufficient facial hair growth.