I agree with Trisia about the "building" example; I would prefer, "This building has been under construction..." as well.
But the "kitten" example seems just fine to me, and I would probably use it more often than not, and I would most likely use it much more often than the version containing "undergoing treatment for an eye infection."
The phrase also seems to be quite common colloquially, as in:
"I'm having trouble with the machine again."
"Yeah, I'm not surprised. That machine has been being difficult the whole morning."
This building has been being constructed for four months and has still not been completed
My five year old kitten has been being treated for an eye infection.
But these sentences does not make any sense to me. Could you explain what they mean?
They make perfect sense! Lets use the example: My room has been painted and My room has been being painted.
In both sentences you can add since or for as it is Present Perfect and Present Perfect Continuous (also known as Progressive).
The first sentence means that your room has been painted- you started in the past and you have just finished. It is done. You are not going to paint anymore as the room looks lovely . Done deal, when the second sentence has a slightly different meaning. You started painting in the past (since yesterday, Monday, July or for two days, 3 months etc.) and you have been painting up till now when you say this sentence butyou havent finished painting meaning you might continue painting tomorrow because it has not been done yet. In both cases you pay attention on the duration. It is very rare to use the Present Perfect Continuous Passive form but it is correct. I hope you understand what I meant.