There are tons of options out there but your probably going to want to stick with a neutral color. Being a former "color expert" from HD my answers are going to involve colors from Glidden and Behr paint. If your interested here are some great colors Cookie Crumb (Glidden), Warm Muffin (Behr), Raffia Cream (Behr), Cozy Cottage (Behr), Oklahoma Wheat (Behr). If you find a color you love, I really recommend using a Latex Flat Enamel. It's washable but not shiny. Satin's and Semi-Glosses are HIGHLY over-rated. Good Luck!
You can accent the walls gray like paint to adjacent walls gray. And accent the room with gray and black accessories. Good Luck.
Try a nice deep champagne beige.. It has suttle pink undertones and will go wonderful with burgandy. You could accent the room with pink tones and navy in pillows and accessories. The room will be warm and inviting and so very cozy.
I would go with a grey color to off set the leather on the furniture.
I have my walls painted honey on the top half, a cherry chair rail and olive on the lower half of the wall.
Peach would bring down the burgundy tones and gray accents like the woodwork/floor boards would give it some contrast color. Pink would also be nice if a light color in the range of the burgundy.
Paint them a neutral colour and it will give you far more scope with other furnishings.
My opinion would be a silvery gray, as to not clash with the burgundy of your furniture. I have to assume you have natural floors, you didn't say. If you have a beige or similar color carpet on your floor, then go with a cream tone on your walls & being in more accent colors with your pillows or curtains..