I don’t have a full installer set up yet for Linux, as it’s the OS I’m least familiar with. That said, you should be able to extract LightBurn to a folder (be sure to extract with folders enabled) and then launch either LightBurn or AppRun. You might need to “chmod +x LightBurn” (shown here: Using Lightburn with linux mint) to make it executable, and add LightBurn to the dialout permission group. (AppRun is just a shortcut to LightBurn).
If any of this trips you up, please let us know so we can further assist.