Our chimney removal took two long days and was really straightforward. This is MESSY work, so we specifically undertook it before we moved anything into the house, and even then, we taped off the room to minimize the dust. This is do-able if you already live in your house, but you're going to want to move everything out of the room and tape the doorways off like whoa.
PS don't wear Toms while you're demoing your house. I am an idiot and did so long enough to sustain two nail-in-the-foot injuries + a call to the doctor to check on my tetanus shot status, so don't be dumb like me. Also, maybe wear your safety googles over your eyes instead of on top of your head? Just a suggestion.
The chimney structure that extends to the basement floor is basically just there to support the bricks above it, so that was next to go. About 100 years of old ash and roofing materials were hiding in that shaft, so the basement-level removal was especially messy. Two days after we'd gotten the keys to the house, our chimney was gonzo:
Because I know pretty pictures are the best part of every before/after, this is what that former fireplace corner looked like later in the summer of 2013, after we'd moved in.