I am certain you can, but, it's like a cell phone, if you leave it on, it will take hours more to charge. Frankly at night, when I charge my cell phone, I don't want to leave it on. Voice mails will be available in the morning, and my phone will be 100% charged, instead of the 80% that it will charge overnight, if I leave it on.
Can you use Amazon's replacement ac adapted to recharge your Kindle Fire? I left myrehular charger that came with my Kindle Fire at my daughter's house which is too far to go pick it up today. I purchased the Amazon. AC Adapted with the USB cord awhile back but due to some problems I have experienced with thlithiam battery I am now Leary of using anything other than the charger that came with the Kindle Fire. I can't seem to find the product info on the replacement charger and am concerned it won't be comparable.
We bought our Kindle Fire in April and encountered the same problem. Bought a new adapter and it did nothing. Spoke with customer service a few weeks ago and they sent us a completely new Kindle (we just had to give a credit card number to ensure we mailed back the defective one).
A lot of chargers block use while charging. The charger that came with your dock allows surfing while charging because it was specifically made that way. The FireStation I mentioned above also allows use while charging - again, it was specifically designed that way for use with the Kindle Fire.