The simplest answer to the problem of people using their phones to look up the answers is to have the quiz somewhere with no mobile signal (or at least below 3G) and turn off the WiFi. Sadly that’s not a practical solution – Faraday cage around the pub anyone!
You need to make it clear that anyone caught using their phone can carry on with the quiz if they want, but won’t be included in the results. Try to move around so that you can see all of the teams at some point during every round, and if you see someone with their phone out start paying them extra attention.
However, at a recent quiz one couple had brought along a teenager too young to be left at home, who had no interest in the quiz. There was absolutely no way said teenager wasn’t going to be using text, WhatsApp, Facebook, Instagram et al, which looked exceedingly suspicious to all the other teams. However, give this team an ultimatum that the phone goes off or they’re excluded, and one way or another someone on that table’s unhappy, and you’re a team down next time.