"Off the menu" isn't an idiom or phrase. These are both creative inventions of headline writers trying to be clever. It's coincidence that they both came up with the same phrase. Headlines often try to be clever and make little jokes or puns. In the first story: A restaurant menu is a list of possible choices. Diageo owned the Burger King hamburger chain. It was trying to sell it. It can't. It has given up. Selling Burger King is no longer a possible choice. It is "off the menu." The headline writer thought it was clever to use the word "menu" because Burger King is a restaurant. If some big company owned an airline, was trying to sell it, and had given up, the headline writer might have written "Sale of Frammis Airways no longer on Megacorp's flight plan."

The first link, about Burger King, is speaking about how the Burger King franchise is no longer for sale ('removed' from the menu) and is mostly talking about investments and competition in the stock market. The second link references people going to McDonalds but going there for reasons completely unrelated to McDonalds itself (not related to the menu). The title: "Many Off-the-Menu Sales" is a reference to drug sales. "Off the menu" generally refers to something no longer available, usually used in the food industry when certain food items have been removed for some reason or another (food products no longer being accessible, previous food poisoning in a restaurant, or other events). It could also be used when there are multiple options, and one of the previous options is no longer available, such as a car or bike at a dealership.