Origin
This belief is an ancient one — our oldest print sighting so far dates it to May 1960 when it appeared as the plot of one of the "Gus And The Model Garage" series run in Popular Science.
Unlike the venerable "purse hung from a car's choke knob causes damage to the automobile" legend, there is a bit of something to the caution against having too many keys on one's key fob. Say Tom and Ray Magliozzi (Click and Clack of Car Talk fame), "Actually, the answer is that excessive weight CAN damage the ignition switch. We see it most often on Volkswagens, for some reason. But it happens on other cars, too."
I have friends that tell me that the numerous key rings and good luck charms I carry on my car keys, because of it's weight it will damage my car's ignition. Then others that tell me its nonsense. Will the extra weight on my key ring damage my car's ignition? I sure would appreciate your response.
The Magliozzi brothers are quick to point out that what constitutes "excessive weight" on a key ring puts the danger point well past what most folks routinely tote about with them. "If you have seven or eight keys on your key ring, you're not going to do any damage. The ignition switch can handle that. But if you carry around 20 keys, that's a fair amount of weight. That constant downward tug on the ignition switch can cause it to wear out prematurely."
Sources
Bunn, Martin. "Gus Licks a Weighty Problem. Popular Science. May 1960.
Magliozzi, Tom and Ray. "Click and Clack: Car Talk." Buffalo News. 28 March 2006 (p. C9).
Paul, John. "Car Doctor - Heavy Key Ring Could Be Culprit in Malfunctioning Brake Light." The Providence Journal. 27 October 2007 (p. C11).
Tallant, Nicola. "More Keys Than a Mountjoy Warder." The Mirror. 4 July 1998 (p. 6).