According to this story, Mitt Romney received zero votes in nine precincts in Cleveland, Ohio. In seven of those precincts, though, some minor party or independent presidential candidates received votes. Thanks to Jim Goodluck for the link.
Also, in Hill County, Montana, precinct #28, which seems to be entirely or almost entirely on a Chippewa Cree reservation, the vote for President was: Obama 552, Gary Johnson 8, Mitt Romney 2. The U.S. Senate results in the same precinct are: the Democratic nominee received 551; the Libertarian nominee received 5; the Republican nominee received 2. Thanks to Mike Fellows for that news.
It would be interesting to see if there are any precincts that are composed of all Latter-Day Saint members (perhaps in Utah, Idaho, Wyoming, Arizona, or Nevada) that were unanimous for Romney. Of course not all LDS members are Republicans.