Yes, birds do need to eat nuts, seeds, etc. It's what they eat in nature, and it serves their GI Tracts the entire way through, along with their beaks, and their mental stimulation. So healthier nuts as treats are good, but peanuts present a couple of issues...
The main one is Aspergilliosis, as Kiwi already mentioned, and that's the peanuts in the shells. So any seed-mixes that have in-shell peanuts should be avoided. Other than that, peanuts have a very high fat content, though they do have also a high protein content...So giving a raw, unsalted peanut out of the shell as a treat once in a while is okay, but not on a regular basis, and as Gail said, there are much healthier, safer nuts to give your bird, so peanuts really aren't needed.
Almonds, Walnuts, Brazil Nuts, Pecans, Filberts...all much healthier and safer than peanuts, but again, use them as treats, not as a part of his "diet", which needs to consist of a healthy pellet supplemented with a healthy, varied seed-mix, and then lots of fresh veggies and a little fresh fruit as a treat (too much sugar is not good either)...
Generally though, to answer your question, when you hear about peanuts being bad, it's the peanuts-in-the-shell and the risk of Aspergilliosis that is the main thing people are talking about, as Kiwi stated already...