Getting started
Beyond These Stars: Can You Continue a Demo Save in the Full Game?
The quick answer
No. Demo saves are incompatible with the full game, so you need to start a new save in the full release.
Start a new full-game save
Open the full version of Beyond These Stars and begin a new game. Your demo settlement cannot be continued there as a compatible save.
You can keep your demo save as a record of that playthrough. Deleting it is not required to begin again.
Avoid forcing a conversion
Renaming or copying a demo save does not make its data compatible. Do not overwrite your original files in an attempt to force the full game to load them.
This concerns the demo-to-full-game transition; it does not describe compatibility between every future full-game patch.