A nice "feature" of Godot is that it doesn't require any runtime to run (such as the Visual C++ redistributable or DirectX, both of which are commonly required by games created using other engines). This is valid both for the editor and exported projects.
Note that if you use Mono in your project, you will have to make sure the Mono DLLs are located besides the exported project binary (and distribute them alongside, e.g. in a ZIP archive).
Linux binaries are statically-linked for the most part and should be able to run on "fresh" Linux installations. However, if you've compiled Linux export templates yourself, they will only be able to run on distributions that are as recent or newer than the distribution the binaries were compiled on. This is why it's recommended to build Linux binaries on the oldest still-supported Ubuntu LTS version, which is Ubuntu 14.04 as of writing. Official export templates follow this recommendation and should be able to run on most Linux distributions, LTS or not.