Watch this guy - he's hillareous, and he knows how to tape a rim. It's the tap, as previously said.
Tape: Throw away anything tape-ish that you have that does not say 'Scotch 8898' on it. Same tape the guy in the video uses. Use at least TWO wraps. Tape width should be the same+2mm wider than your rim internal width.
You'll get there!
Oh, and the reason the tires are demounting is because the tape at the outer channels is not thick enough and the new tires have a smaller bead than the old ones. It's the tape...