The chimney is on an inside wall between two rooms.

Short answer is Yes. Masonry fireplaces require a massive footing to support all that weight. With the amount of masonry you plan on removing you'll be lessening a good portion of that weight. Nevertheless, a structural engineer will need to figure a solution to support the remaining portion of the chimney prior to removing the portion below. That might require structural support in the area where you plan to remove the lower portion. Defeats the purpose. With that said, its my experience that removing the entire chimney, patching the floor, ceiling and roof is the simplest and less expensive alternative. Honestly, I wouldn't want a "top half portion" of chimney just "hanging around" in my house.

Question. The amount that you would like to remove... Is that 25%, 50%, 75%...?

Can't imagine leaving that much weight hanging about! Take it all down and never worry about it again.

It would be 50% of the chimney removed.