For restaurant owners: ease of updating the menu (pricing, dishes, suggestions,...), (perceived) lower cost of maintaining paper menu's (whether disposable or reusable).
For patrons: you can bookmark the menu if it's a webpage on the restaurant's website, there's (perceived) accessibility (e.g. font zooming) but that's offset with the higher bar of scanning a QR code.
Maybe there's a niche market to be spotted for BI / data acquisition? It would be quite a tenuous proposition, since consulting a menu isn't the sames as capturing orders. Not to mention the hairy privacy issues associated here.
Here's an alternate business case where the perceived benefits of QR codes are greatly diminished pending the specific context: museums and galleries. The success of using QR codes works pending on the type of audience, the type of collection, the vision on the specific user experience a museum wants to achieve, the ability to spend time / resources on providing and keeping digital content up to date, time spend helping people out,... It's not as easy as "slap QR codes on a wall, and they will be scanned."
In the same vain, QR codes may work well in some venues, and they won't work at all in others. Context matters. A lot. For instance, if you operate a bar or restaurant in a cozy, 19th century, cottage setting and you want to foster that specific atmosphere / experience towards your patrons, sticking QR codes to the old wood would immediately detract from that. I don't see QR codes work greatly either if your into a gastronomic / gourmet niche for instance. Opposite of the spectrum, there's fast food chains and loads of small businesses who just want to get food out of the door and not much more: to them, a menu is just a business expense and a QR code is a good solution to cut costs.
I don't really mind QR codes. When I go to a run-of-the-mill place for take out, I don't mind scanning a code. My expectations will change, however, if I'm going to spend time and money hoping to get a specific experience out of it.