Renny's Dining Rooms seem to have disappeared with no other trace on the Internet, and while it had been clearly at No 41, I can't recall exactly where it stood. But I think from the name it was probably on Rennie St, just a little to the west of the Blackfriars Rd, possibly close to its start on Upper Ground, but no trace of it remains. But the whole area had a strong connection to the Rennie family.
The street will have got its name from the engineers John Rennie the Elder and John Rennie the Younger who had their engineering works in nearby Holland St, but lived at 27 Stamford St, a few yards from where Rennie St crosses. They were the best-known civil engineers of their era, with the elder responsible for many canals and other great schemes, including many bridges. The old Waterloo Bridge was perhaps the best known by the father, while his son built London Bridge, which was sold in 1967 and some of its stones used to build a replica in Arizona. There was also Rennies Wharf nearby.