AppImages are portable software that do not require root permission to be run. They can be run without root permission:
chmod +x ExamplePackage.AppImage
Most AppImages require sys-fs/fuse on slot 0:
dlopen(): error loading libfuse.so.2 AppImages require FUSE to run. You might still be able to extract the contents of this AppImage if you run it with the --appimage-extract option. See https://github.com/AppImage/AppImageKit/wiki/FUSE for more information
This error can be solved by installing sys-fs/fuse in slot 0. Slot 0 has version 2 of fuse required by most AppImages. Whereas sys-fs/fuse in slot 3 which has version 3 of fuse. The highest version available is installed by default, currently in slot 3, if no slot is specified and doesn't solve the dependency problem.
emerge --ask sys-fs/fuse:0