I have to say I researched this topic a lot as I keep and breed many kinds of birds as a hobby as well as my chickens. I found many books and people that would recommend feeding dairy products like milk, cheese and live yogurt (for good gut bacteria) to pet birds for extra protein and calcium.
I have fed my birds these things in the past with no ill effects. However, I have decided to stop!
I realised, while it is not harmful in the short term, it may be bad long term. This is because, birds have not evolved to eat dairy products - its a totally unnatural food for them to eat. Their bodies digestive tract is not designed, or capable, or digesting diary products properly.
Lactose is a sugar in diary products, and no birds have the enzymes to break it down and digest it. That's also why the beneficial bacteria in live yogurt is no use to the bird!
I would think a bird that is regularly fed dairy products will eventually get some ill effect on its digestion, such as is see in humans who are lactose intolerant - diarrhea, abdominal pain and lethargy.
It is also totally unnecessary to feed dairy products, as we can feed other foods to the birds for protein and calcium. Dairy produces has also a high risk of going bad if not eaten by the bird after an hour or to, and may develop harmful bacteria on them which will make the bird sick.