A user recently reported being unable to open a .mobi file after downloading it using the Kindle Fire. After some investigating I think I found the solution.
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 this solution does not work for anyone, please post here to let us know. Include any additional details. If the solution does work, feel free to post a confirmation here too so that we can be sure this solution works. Thanks!