Try a wet rag covering the tip of a flexible putty knife. May take a
bit of effort. Depending on how much paint you need to remove, you'll
have to move the rag around to clean spots.

To avoid this I use the putty knife -- either the end or the side -- to
push down on the edge of the tape. This will seal the tape against the
woodwork.

It also help if you avoid globs of paint on the tape [not saying that's
what you did, but...]