A national survey by Public Policy Polling found that 48% of Republicans (and 26% of Americans generally) endorsed the unsupported smear that President Obama doesn't love America (27% of Republicans said Obama does love America and 25% were not sure). Those numbers are even worse than the myth that Obama wasn't born in this country, which was endorsed by 42% of Republicans (and 23% of Americans generally) in a September PPP poll.
Update 10/23 2:05 PM: As a point of comparison (per Jinchi's comment), a Fox News poll in June 2008 asked "How much do you think Barack Obama loves America?" (rather than "Do you think that Barack Obama loves America?"). 27% of Republicans said "a great deal," 34% said "somewhat," 14% said "not much," 12% said "not at all," and 14% said they didn't know. Though the question and response options varied slightly, those responses are substantially more positive than those found by PPP.
Update 10/25 8:42 PM: Per MartyB's comment, it's worth clarifying that the reason I compared the two posts above. While the two claims obviously differ in terms of the extent to which they can be disproven, both polls demonstrate that Obama is not viewed as a legitimate president by much of the GOP base.