If an error in the module causes configuration settings or other data in your database, they will remain even after removing the module. So yes, it can cause problems for other Drupal modules or core even after removal.
Please note that this can happen with โstableโ modules, as the decision about when to declare the module โready for releaseโ more or less corresponds to the support!
This is one of many reasons to never, never test new modules (stable or not) directly on a production installation. First create a test installation, install a new module (s) and test, test, test.
The same applies to updates, even for Drupal kernel updates. Always run a test run with a separate installation first (at least make sure you have a working, restoreable backup of your database and code that you can return to in case something goes wrong).
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