TLDR: Nope! The testing domain in Memberstack has no effect on your finished site.
Let's say your site has two different domains: one for testing and another for your "actual" domain.
You do NOT need to update the testing domain. Memberstack asks for a testing domain so its validation bots can find your site and help you avoid mistakes.
You could completely remove the domain from app.memberstack.io and your site would continue to function.