Therefore bad covers make bad books.
However it is original cover it should be left on, no matter how ripped, folded, or otherwise damaged it is.
Bad covers - especially photographs of a book with bad covers- reduces the value of a book-with the exception of above. However you should keep the cover on until you sell it to protect it. So in summary-remove bad cover, photograph it, put bad cover back on until you sell it.
Obviously the book won't keep so long for the seller as it would do with the bad cover on-but then again you're giving the book you photographed, the seller wants a book that's protected in the post. You could of course offer the cover to them, as long as you say so. It may seem rather contrary but a temporary good product for some reason is better to sell than an average product that lasts longer.
On the flip side, you can bargain with sellers who offer tattily covered books.
Another point is that ex-library books have plastic covers. Should you buy a lot of these, you can take the covers off and sell them.