It gets better than that, the RIAA has tried to muscle bars for RIAA "protection" fees for the songs played on tv commercials! If a bar has a tv (with audio) on, they could be strong armed into paying for RIAA license fees.
What is ridiculous is that the sponsor company (Burger King for example) has already paid for the licensing of that song for the ad.
Cover bands get a free ride from the bars' RIAA agreements but I've even heard of RIAA goons aproaching the bands for a cut of the receipts of the night's take.
At least until recently, copyright law recognized the concept of "performance rights" for musical and textual compositions, but specifically not for audiovisual transcriptions. It still recognized the concept of "duplication rights" for audiovisual transcriptions, but not performance rights.
Unless things have changed, the RIAA has no legal basis for demanding royalty payments. ASCAP/BMI might conceivably have a legal basis, except that the short musical clips heard in commercials generally either already have had license fees paid by the advertisers, or are of a nature that they could be considered "fair use".