Now I don;t use Gnome, just as people who know me also know I haven't used Windows since the 1990's.
But I have had video wallpaper and web browsers as wallpaper for years, and reading posts that say it is not possible, or talk about writing shell scripts that take photos of your browser and set them as a wallpaper slide show annoy me.
So here is the easy way. I downloaded Gnome to test this and it does not work because Gnome has no option to 'keep a window Below Others, but in KDE it is dead easy and I suspect the others will also let you do this.
Ok, so in KDE all you do is:
1 - Right click the title bar of your browser.
2 - Choose 'More Actions'
3 - Choose 'Keep Below Others'
4 - Choose 'More Actions' again
5 - Choose 'Fullscreen'
6 - Read the warning that tells you you probably won't get back out of full screen.
7 - Take note of the 'ALT+F3' shortcut to get out of full screen.
8 - Click ok.
9 - See point 6. Make a text file in /home to remind you.
To get out of Full Screen use ALT+F3 Choose 'More Actions' then get rid of the 'tick' next to 'Fullscreen'. You might want to get rid of 'Keep below others' as well.
I have been using this method to set VLC to play videos as wallpaper for years, but now I see it has a built in "Wallpaper Mode".
It also works for other programs, but can be annoying when you accidentally erase the document you were working on in your 'Desktop Wallpaper Database'.
One program that does make good Interactive Wallpaper using this method is Google Earth. Just go to View and set the Sidebar to OFF.