...at all? Like if the character in question was completely removed, the show would not be affected at all.

Randall Carver and Jeff Conway's "Taxi" characters.

Maybe Ben on the Waltons since he is my least favorite sibling.

The two ladies living in a trailer on Tracker
Bob on The Middle- Chris Kattan started out in the opening credits but was later removed.

Betty Jo and Steve's second child on Petticoat Junction. It was "conceived" as a way to keep Dr. Craig from moving to New York. The final season of the show there was no mention of the baby.