Do it on a busy travel day and suddenly the authorities have to deal with a ton of broken passports. Either they arrest everyone, or they let everyone through.
Of course, you probably couldn't get a suitcase-sized RFID zapper through security, but you could show up, set it off a few times, and then leave without arousing TOO much suspicion.
Or at the passport issuing office. Now that would be chaos.
That was in the days before RFID. My passport has a bar-code and every entrance station already has a reader, so RFID is redundant anyway.
The tourists and kiwis with old passports all line up in a giant queue, while myself and a few others walk up to some machines, scan our passports, have a quick photo taken, and then just stroll on in!
I've flown to NZ with just carry-on luggage and caught a cab within 10 minutes of the plane doors being opened.
So, yes, they do track you.
(FTR, my passport is pre-RFID.)
Just about any passport issued in the past decade or so has a machine-readable text strip on the photograph page. When the passport control officers take your passport they don't just compare your passport photo to your face -- they swipe the passport.
The RFID chip just allows them to store more data and get at it faster than the older machine-readable passports. The information that your passport (and its bearer) have arrived in country X is still available to country X's immigration database the moment you're past the checkpoint.
(Other risks of RFID passports include tracking or cloning, if the owner isn't keeping it in a tinfoil wallet or if the issuing agency's private key gets leaked. See also discussions on comp.risks passim.)