Over the past two years Comedy Central’s Jon Stewart has poked fun at fast-food chain Arby’s dozens of times in segments on the “Daily Show,” routinely linking the brand to cringe-worthy topics from Ebola to digestion problems.

His on-air quips have included: “Arby’s, for when you’re wondering what it tastes like when a cow dies” and “Arby’s, the meal that is a dare for your colon.”

But...