The answer is a resounding "yes and no." It's hard to separate medical truth in this matter from the overlap with the "government monitoring/New World Order/black helicopters/Mark of the Beast" paranoid conspiracy theorists who have taken to the internet like ducks to water.  And nothing scares people right out of their rational minds like saying that something causes cancer.

Microchips, because they are injected, can cause postinjection fibrosarcoma.  This is a form of cancer which can appear at the injection site of a dog, cat, or ferret after ANY injection, even a simple blood draw.  Postinjection sarcoma is vanishingly rare, and that the benefits of whatever's being injected far outweigh the risks.  (You can tell how rare it is, because when was the last time you heard of a case of it?  I'm guessing never.  And how many pets do your friends and family and coworkers have, and how many injections have those pets been given?  I'm guessing thousands.)

Here's the best, seemingly most authoritative article that I found online: Fibrosarcoma with Typical Features of Postinjection Sarcoma at Site of Microchip Implant in a Dog: Histologic and Immunohistochemical Study

 It follows the case of a French bulldog who presented with a lump at the microchip injection site.  The final analysis was postinjection fibrosarcoma - which can occur with any type of injection.  The presenting veterinarian in the end wasn't able to definitively state whether the microchip caused the sarcoma, or any of the other vaccinations the dog had been given.

"Microchips cause cancer" is a bit misleading and overstated, but it's not UNtrue.  But it's rare enough, and the potential benefits of microchips are so phenomenal, that I think it's definitely worth the risk.  Microchips absolutely save pets' lives every day - the first thing any shelter or vet does when they receive a stray animal is scan it.  Microchipped animals get an extended lease on life at the shelter, even when the owners can't be contacted.  

I remember a throwaway scene in a news story about Hurricane Andrew.  Huge storage facility filled with sopping wet rescued dogs and cats stacked floor to ceiling in crates and boxes and anything that would hold them.  An SPCA tech was moving down the line, scanning animals for chips.  He scanned a cat, the scanner went "Bleep," he picked up the cat and said "You're going home!" and went off to contact the owners.  

All the rest of the dogs and cats - the ones without microchips - were destined to be shipped to an animal control three states away, because Florida's shelters were completely filled.  Three states away, in the aftermath of a hurricane, and their owners had 72 hours to locate them before their pets would be euthanized.  Tick tock!

Made an impression on me, it did.  I had been uncertain about microchipping, but the next time I went to the vet I had my dog chipped.  I knew which batch I wanted him to be in, if something like that ever happened.