I think there are two ways to look at this. On the one hand the neighbour below is losing the future option of having an open fireplace. On the other it gives them the opportunity to remove their chimney breast at a later date, which is quite a common thing to do in flats to gain more space.
Anyhow, it is precisely for this reason that I say it's essential you get legal agreement with interested parties. If you were to just proceed you might find yourself having to reinstate it later when the neighbour decides to have an open fire and discovers that you've removed the chimney (which is a common part) without authorisation.
My downstairs neighbours are really up for it. they wanted to get theirs taken out ages ago but wanted upstairs to do it at the same time. The person that owned this before me didn't want to do it because of the cost.
My neighbours are gagging for me to get rid of mine as it will mean less hassle when they get theirs taken out. Will diefinitely discuss it properly with them though and get an agreement in writing.