If you can tolerate the paint remaing in the pits of the surface you can remove it.
Latex can usually be removed with a scrub brush and TSP, 409, or even Spic and Span.
Oil based paint is harder to remove and will usually require paint stripper (and this may alter the brick color).
Pressure washing and sand blasting are pretty much non-starters in the finished areas of a house.
Do not use any brushes with plain steel bristles. It can embed in the brick surface and cause rust stains to appear later.
A couple layers of 6 mil plastic and canvas drop cloths on top of the plastic will help protect the floor.
It is hard messy work. You can always give it a try and if you find it to difficult or do not like the results just apply another coat of paint.

I have read that Peel Away works. It's a stripper that you put on like frosting, cover with paper, let it set and work, then pull it off; ideally the paint comes with it. Formulas 1 and 7 are supposed to work on brick. It's expensive but you can get just a quart to do a test area. I used PA 7 on some window trim; it did a good job on the newer layers but didn't have much effect on the deepest layer, which was the one I was really trying to remove (lead paint).

I now have a painted brick fireplace (different house) that would look better stripped, so am interested to hear what results you have.

Don't give up on sand blasting. I've seen it done successfully on interior brick work. You just need to find an experienced professional sand blasting company. The surrounding areas can be protected from damage.

I've used PeelAway1, Ace Hardware concrete cleaner, assorted brushes, rags, water and alot of elbow grease. It got the brick back to like new. Once we got all the paint and 98% of the residue off, wiped the entire surface with a 50/50 mix of linseed oil and mineral spirits. It turned out wonderfully!

i would skip the whole paint removal and repaint it with brick-anew-it is cheap and looks like brick.
www.brick-anew.com
mw

Here is a link that might be useful: brick-anew