I could only find fairly old literature on the subject, so take this with caution.
Some papers seem to indicate some sort of genetic component to tongue rolling, although the precise molecular basis are not really described (but, again, these are 30-40 years old papers...)
From: Inheritance of ear wax types, ear lobe attachment and tongue rolling ability. - Cruz-Gonzalez and Lisker - Acta Anthropogenet., 1982
The mode of inheritance of ear wax type, ear lobe attachment and tongue rolling ability were studied in 77 families with a total of 293 children. The results clearly showed that the dry ear wax type and the attached ear lobe type represent the homozygous state for two pairs of autosomal recessive genes. The evidence for the same being true regarding the lack of ability to roll the tongue was less conclusive in our material, but this could be due to difficulties in communication between the examined individuals and the examiners.
Tongue-rolling phenotypes and geographical variation in the United Kingdom
The distribution of tongue-rolling phenotypes in a sample (n = 477) of undergraduate students of the University College of Swansea (U.K.) was studied. The birthplaces of these students were also recorded. England was divided into six areas, and Wales was left as an area on its own. The data suggest that those students who come from north-east are more non-rollers, which may be due to mixture with Scandinavians.
Left-handedness and tongue-rolling ability.
948 undergraduates at The Ohio State University were administered the 10-item Edinburgh Handedness Inventory and asked to indicate the extent to which they could turn up the sides of their tongues. Significantly fewer left-handers than right-handers (62.8% and 74.8%, respectively) reported being able to turn up either or both sides. Sex differences in tongue-rolling ability were also noted. Among the 403 men included in the final sample, 77.4% could roll their tongues, whereas only 69.7% of the 491 women could do so.
This paper, however says (luckily Google finds a PDF here... seriously it should be illegal to have papers from the 70s behind paywalls... but I digress):
No evidence for a genetic basis of tongue rolling or hand clasping - Martin, J Hered., 1975
OCCASIONALLY one still finds in elementary genetics courses and elsewhere, the ability to roll the tongue and the manner of clasping the hands cited as examples of simply inherited human polymorphisms. This is despite extensive evidence to the contrary that has accumulated over several decades. Sturtevant T found that about 65 percent of persons studied were able to tum 1.JP the edges of the tongue and although some people were able to learn the ability, family data indicated that it was inherited as a simple dominant allele. However, Matlock reported 7 out of 33 monozygotic (MZ) twin pairs discordant for the ability, and Sturtevant concluded that there is sufficient nongenetic inftuence to make the character practically useless as a genetic marker. He confessed to being "still embarrassed to see it listed in some current works as an established Mendelian case."
The paper is also cited in the "Myths of Human Genetics" page of the University of Delaware.
So, all in all, it seems that there is evidence against a Mendelian transmission of this character, although these studies are quite old and some further analysis is warranted to exclude non Mendelian (read epigenetic) transmission of tongue rolling.