Eames chair - burlap cleaning or replacing?  

These were first upholstered in nylon hopsak designed by Alexander Girard. Herman Miller still offers Girard hopsak but now it's recycled polyester--same weave, same colors, just a different fiber content. (Hopsak refers to the weave pattern. Burlap is a utility fabric made of hemp.)

I've never cleaned one of these chairs specifically, but I've read that water-based cleaners are the way to go. I've gotten good results cleaning wool upholstery with just Dawn detergent suds dabbed on, blotted, then repeated as needed. It's important to rinse the fabric afterward, which I did with clear water sponged on and blotted, then vacuumed out with a Wet-Vac to get it as dry as possible.

Is the foam in good condition? Early Eames chairs were done in latex foam which gets dry and crunchy after a few decades.

I think Spanky is right about the foam. I just finished a project bringing 8 of these back to life - mine are DAT bases, but same vinyl and hopsack upholstery. The foam was hard and gross. It's a bit of a pain, but ripping the upholstery off, peeling the adhesive and cleaning and reconditioning the fiberglass will have pretty good results. If you go that route, try to keep the upholstery intact when you take it off so you could make a cushion with the bottom section (the hopsack section) to use as a pattern. You'll want a cushion as this version of the fiberglass chairs have airholes and are drilled through for the base to connect.

Thank you very much, spanky and frankielemmons, for your answers and suggestions. They are greatly appreciated.