According to most official documents, any passport which has a faulty rfid will be assumed tampered with, and will become invalid. So unless you want to be refused entry and forced to replace your passport, this is unadvisable.
You could cause even more of a hassle then by building an RFID zapper (http://en.wikipedia.org/wiki/RFID_Zapper). A powerful one that fits inside a suitcase could be set off in an airport concourse, frying a large number of people's chips without them ever knowing (or any damage to the passport itself occurring).
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.
I've had border guards order me to replace a passport that I carried with me in a rainstorm. It was perfectly readable, but they didn't like the way it had visible damage. Maybe the "security" features were less checkable.
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.
I really like the RFID entrance I get when traveling back to New Zealand... especially after a 13 hour plane ride!
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.
It's certainly convenient, but then the authorities can track where you were/where you're going. I don't know if they do the same when manually checking the passport, but I'll assume they don't?
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.)
I've been to Peru a few times in the past couple years. Every time I passed through the immigration counter, they swiped my passport through a reader, and up on the customs officer's screen came a list of my travels since getting my current passport. Yeah, Peru sees a lot of tourism traffic -- more than many countries, but they're also not exactly first world, and technology penetration is spotty, at best.
They definitely do; that's the whole reason why you are required to have a passport in the first place. RFID just makes the process more efficient; there's nothing on the chip that isn't contained within the combination of your passport number and their database.