The terms for the Apple AppStore have been shown to conflict with the GNU GPL in such a way as to prevent GPL apps from being in the store — see this question. Are the Amazon AppStore (the standard repository for software for FireOS devices) terms similarly restrictive?
TL/DR: The Amazon AppStore is compatible with GPL software, as long as it stays away from the DRM features.
The terms of the Amazon AppStore, which can be found here, include some restrictions that would be incompatible with the GPL or other copyleft licenses, but most of those restrictions can be waived by the publisher of an App.
3.3 Separate Licenses to Use Apps.
Your use of an App will be governed by the terms and conditions of an end user license agreement between you and the Publisher of the App (a "EULA"). The EULA for an App includes the terms set forth below (the "Standard EULA") and any additional end user license terms included in the App (the "Publisher EULA"). [...] As used in this Section 3.3, the term "App" does not include any Appstore Software included within an App (your use of Appstore Software is instead governed by Sections 3.1 - 3.2 above). The Standard EULA includes the following end user license terms, and if the App does not include a Publisher EULA, these terms will constitute the entire EULA between you and the Publisher:
(ii) If the App does not include a Publisher EULA that specifies App license rights, Publisher grants you a limited, nontransferable license to download and use the App only for your personal and noncommercial purposes.
(v) You may not modify, reverse engineer, decompile or disassemble the App in whole or in part, or create any derivative works from or sublicense any rights in the App, unless otherwise expressly authorized in writing by Publisher.
3.1 Appstore Software.
We may make available to you software that you need to install on your device in order to download Apps purchased from the Amazon Appstore to that device and to access or use Apps on that device (the "Appstore Software"). [...] Publishers may choose to protect their Apps using a digital rights management ("DRM") system implemented through the Appstore Software. Appstore Software includes any software we include in an App to enable the DRM or for other purposes. If you uninstall the Appstore Software from your device or sign out of your Amazon.com account in the Appstore Software, you will not be able to access or use any Apps on your device that Publishers have protected with DRM. You may not attempt to disable, bypass, modify, defeat, or otherwise circumvent the DRM or any other security or content protection system used in connection with the Amazon Appstore.
- You can distribute GPL-licensed software via the Amazon AppStore if you accompany that software by a Publisher EULA that specifies that the GPL license applies and explicitly mentions that you are allowed to make derivative works, modifications, do reverse engineering, etc. according to the terms of the GPL license.
- You can not distribute a GPL-licensed app that uses the AppStore DRM system, because the DRM system does put additional restrictions on you as a user that are incompatible with the GPL license.