Sometimes tape is necessary....
The problem you describe is probably not the tape, it is the paint. This is usually not a problem with flat paints, but paints that have a sheen or paints with an extended "dry" time can cause a problem.
We "back cut" trim, that is, we paint all the windows doors and trim first ( except baseboards ) then we paint the walls.. truing all the trim edges/lines as we cut in with the wall paint... makes for a much easier hand cut...
As far as the type of tape, 3M's blue "Delicate" seems to work the best.
Best of Luck
rmichael pro painter