How to Sideload Mobi books without Restarting
Install the File Expert app first.
1. Add the mobi books to the Books or Documents folder.
2. Open File Expert and navigate to the folder you just put your books in.
3. Touch a book then hit Choose Applications from the options that pop up.
Note: The first time it will give you a list of apps to pick from. Check Use by default box at the bottom and choose the Kindle app to open the book.
The next time you open a book in File Expert you'll only have to touch Choose Applications and it will auto open with the Kindle app.
4. Do this for each book you've sideloaded. If you sideload via USB the books should appear after you hit disconnect from your PC and there's no need for File Expert but if you use Dropbox or another downloading method. This is the only way I know without having to restart the Fire.
Note: The Kindle app wont open when you touch a book. it opens in the background where you can't see it but it should have worked.
Now go check out the Books or Documents shelves and you will now see the books you've just side loaded appear!
Credit goes to SCION.