I've got a few hardcover books. I usually read paperback, but sometimes I'll find myself with a hardcover book.
Often, these will have jackets on them. Paper jackets that rip and get in the way and fall off when you try to move them. You can't even use them as proper bookmarks, they'll just slip out. (No, I love those covers, why do you ask?)
Is there any harm that it will do to the book itself if I remove the jacket and permanently keep it off?
I don't believe dustjackets protect a book to a significant degree. But I am in favour of keeping them with the book.
Dustjackets are often prettier than the book.
The flaps of a dustjacket can be used as bookmarks (unless the book is very big).
The dustjacket can be removed while the book is being read and left on the bookshelf as a marker of where it goes.
This article from 2015 argues for keeping dustjackets on the books - for decoration, for keeping the actual book undamaged, and for the extra information printed on the inside flaps.
But my favourite use of dustjackets (also mentioned at the end of the above article) is for disguising the book I am actually reading.
The dust jacket is more delicate than the stiff covers of a hardcover book, so it does not really need to be around the book while you're reading it. In fact, that is a good argument for temporarily removing the book jacket while you're reading the book. As Biblio.com points out (emphasis added),
The dust jacket is both the most decorative part of a book, and the most delicate. A missing dust jacket, or a dust jacket that is in poor condition, can cut a collectible book's value more than 50%, and make it harder to find a buyer.
Similarly, Skinner Inc points out,
If there is one single thing that is a make or break for book value, it would be the dust jacket. The value of a first edition copy of F. Scott Fitzgerald’s Tender is the Night with dust jacket is around $6,000. Without a dust jacket….$300. This huge difference in value is largely due to the fact that more than 90% of dust jackets are destroyed, either deliberately or due to their ephemeral and fragile nature. If you have one on a good book, treasure it and be sure to protect it with a plastic sleeve.
If you watch TV shows such as the BBC's Antiques Roadshow, you may have noticed that the condition of an old book greatly influences its value. This includes the presence and condition of the dust jacket, if the book originally had one. If the dust jacket is still present, the book should fetch more money in an auction, for example.
Of course, not all books are collectible, at least not when they are new. But even new hardcover books can become collectible later on.
(The aesthetic argument and the information printed on the dust jacket, mentioned in NiceOrc's answer, are also valid, of course.)
