How to make Unity's Dash save the results filtering?

I do not think you can. These seems to be stored in memory only and lost every time the service is restarted.

You could however uninstall scopes that you do not want to use. You could also disable "remote" scopes by going to system settings, privacy, and turning off "Include Online Search Results".

Failing that, you could always remove the .service files from /usr/share/dbus-1/services/

I STRONGLY recommend AGAINST doing that, but it is an option.

You can also file a feature request with the lense/scope maintainer.

I believe the ability to save previous seach results and filters would be done at the lense/scope level and not at a unity level. Though they may not be true.

TL;DR; You can't as of today.