The issue is that the Kindle Fire automatically saves things to the 'Downloads' directory. However, the Kindle Fire uses the Kindle for Android app. So the book needs to be in the 'Books' directory.
You can either connect to a computer to move the files, or download a file manager like ES File Explorer.
Another option that might work is to email the file to your firstname.lastname@example.org email address. Then they show up as Documents.
If the solution does work, feel free to post a confirmation here too so that we can be sure this solution works.
Update Dec 4, 2015 ~ If this solution does not work for anyone, please post here to let us know. We can hopefully help. Please make sure to include more details so we have some information with which to help:
- (1) Did you try the solution above?
(2) Do you get some kind of specific error message?
(3) Have you ever been able to open a mobi file with your Kindle Fire before?