A google search found a page advising storing walnut oil like you would olive oil - in a dark cabinet or cupboard at room temperature or so.
Quote:
Store at 65F to 75F in a dry and odor-free environment for a shelf life of approx. 9 months in unopened containers. Opened containers should be used within 3 to 8 weeks.
http://www.kicgroup.com/walnutoil.htm
You know, instructions like that drive me crazy, or at least the statement that you should only keep it for 3-8 weeks. I have some walnut oil in my fridge that is probably 3 months old. Does it taste as fresh as when I opened it? No. Is it rancid? Certainly not. I say keep it, when you have a recipe that calls for it, taste it and replace it if it tastes bad.
Honestly, we should have a repository for such perishable ingredients that are only sold in quantities too large for a home cook to use in time.