I've got baby fine hair. When I was younger, if I didn't shampoo every day I'd look like I dunked my bangs in olive oil.
(Now that I'm becoming silver-haired, I no longer have that problem. The silver hairs are thicker and curlier - so they can go a day without a wash.)
Go ahead and wash your hair once a day. Just use a mild shampoo and only wash it the once (don't do the whole lather, rinse, repeat routine).
It won't hurt your hair one bit.