Inexpensive in Beaune
We have never been and are heading there in a couple weeks for a family event. Our nights are pretty much taken, so we are interested in dining options earlier in the day (11:00 to 5:00). We love comfortable pubs, casual cafes, local and not touristy. Extra points for great beer since there will be much wine drinking as part of the scheduled events.
Brasserie Carnot at place Carnot seems to fit your bill perfectly. Very casual and decently priced, if not exactly cheap.
A dish or two (such as the steak) fell flat, but the place was throbbing with locals for a late Sunday lunch. I also recall tasting a couple of very refreshing local beers.
Since I'm allergic to hops, I can't report on beer, but we loved the wine bar Part des Anges in Beaune. Very casual atmosphere, affordable wines by the glass, and really delicious food. As well, in the same block as the celebrated Ma Cuisine, is a delicious pizzeria that serves pizzas and pasta.
Second Le Carnot. It is great & they do have beer. Another place to try is Pickwicks behind the church. It's an English type pub. I'm pretty sure they have food. Last one is Crep des Ange. It's a bit further out (just past place Madeline) but has good crepes. If course you can always grab some cheese at Alain Hess & some beer at the Petit Casino & have a picnic at the edge of the vineyards.