In very general terms, a RMH will weigh 2 to 4 tons more or less. If you use the ordinary "bench" style, it will be spread out over a large floor area, and depending on the specifics of your floor joist sizes and orientation you might be able to support it without added reinforcement. You will be safer adding support under the floor joists, and it might be truly necessary. If you are trying to follow codes, you may be required to have continuous masonry or possibly steel support from the ground up no matter how strong your joists are.
If you want to build a "bell" style RMH with a profile similar to the fireplace picture you posted, you will definitely need full masonry support from the ground up; it is both more concentrated in the loading and taller so as to present more danger from uneven settling possibly tipping the heater over.
What sort of foundation do you have for the existing fireplace? What are its dimensions, and what is nearby in the basement?
Yes, if you aren't interested in keeping the existing fireplace, reuse it as part of the mass! Depending on what exactly you have and how much heat you need, you could have the whole unit in the current fireplace footprint or add some in front of it.
I am a man of mystery. Mostly because of this tiny ad: