How can I watch Amazon Prime videos using Firefox on Ubuntu 16.04?

This is actually a problem with enabling DRM content(atleast in the case of newer versions)

So if its an old version see the above accepted answer, and if you have a newer version and still things are not working, this answer is for you.

Firefox disables playing DRM content by default To be able to watch DRM content enable it by Selecting "Play DRM Content" in Preferences -> Content.

Check this out for more info : https://support.mozilla.org/en-US/kb/enable-drm

Note that I tested this on Firefox 53, I am not sure about older versions.

Yes, you can watch Amazon Videos on Firefox but you will have to use Firefox 49 or newer. At the time of writing, the version of Firefox available in the official Ubuntu repositories is currently at 48, so you will have to use Official PPA for Firefox Beta to upgrade your Firefox to version 49:

sudo add-apt-repository ppa:mozillateam/firefox-next
sudo apt update

If you already have Firefox 49 or newer from the Ubuntu repositories without using any PPAs, then you don't need to use the PPA mentioned above unless you would like to try the beta version of Firefox.

Once you have Firefox 49 or newer, open Firefox and navigate to the desired Amazon Video. There will be a ribbon that asks if you want to play DRM content. Alternatively, click on the hamburger menu button and go to "Preferences" > "Content" and check the "Play DRM content" box.

If it does not work even with DRM-content enabled and Widevine Content Decryption Module Always Activate (both are the default for FF 57, 58 and maybe more), try to solve it by installing VLC

sudo apt-get update
sudo apt-get install vlc

Worked for me with FF 57 and FF 58 (Ubuntu 16.04).