Eagles harmonies is a great topic, and a favourite subject of mine.

I think there are 2,3,4 & 5 part harmonies in the Eagles songs.
Don & Glenn do 2 part harmony quite often in the verse of songs.
Don refers to the Eagles 5 part harmony in his intro to Hole in the World, on Farewell 1.

At live shows most of the supporting musicians sing also, either to add depth to the vocal lines or adding some 'ooohs & aaahs' etc,
On studio albums the Eagles can take more than one vocal part each, but on tour they need some extra voices to replicate that sound.

Those of you who are newer 'Borderers' may not be aware of my 'interest' in Will Hollis, but because of it I'm aware that he takes on the role of 5th harmony & can often be seen singing when the other supporting musicians aren't (there are some good videos from HOTE in the US 2013 on YouTube where you can check this out). I guess that he may be singing the parts Don Felder used to sing.

Like some others have said, I'll need to watch & listen again to come up with which songs have 2,3,4 or 5 part harmonies.