Willow contains salicylic acid, from which aspirin is derived. I don't know if koi might be allergic to aspirin or something, but willow branches are used in shrimp tanks sometimes for alleged antibacterial properties (not sure exactly how effective they are, but the point is that it's certainly not toxic to shrimp). Plenty of people also culture live willow branches in tanks (branch cuttings will root, and act as a fairly effective nitrate buster), at least until they branch outgrows the tank (after which it's sometimes planted outside and a new cutting is taken to repeat the process).
The other downside to willow, however, is that it's a very soft wood, and will rot pretty quickly in your tank. Not really what I'd pick for aquascaping purposes—as others have suggested, manzanita or various hardwoods make for better branches.