Hi CatherineJ; I live in Burford, so here's the skinny...!
In Burford: The Lamb Hotel The Baytree Hotel The Carpenter's Arms (gastro pub)
Within 10 miles of Burford: The White Hart, Stow-on-the-Wold (gastro pub - very very good and not too expensive) Lower Slaughter Manor, Lower Slaughter
Another vote for the White Hart @ Stow, also another one on the green in Stow but cant remember the name, think its the Queens Head, they have a great local pint