Adding a thermal mass bench to existing oil furnace?

I've a three story home, with an oil furnace in a corner of the basement; the basement is built out.

I'm wondering WHY CAN'T I re-route the exhaust piping into the living space and build a cob wall / bench and then route it back to the exhaust piping which runs into the chimney.

Hi Parker ; Welcome to Permies!
If I understand correctly, what your hoping to do.  I worry that your oil stove would quit venting properly if you change its path.  
The super high heat of a rocket stove allows it to push through many feet of  piping.  
Your oil stove needs a path of least resistance.
Changing it could have bad results.

Does your house have a 6" or bigger chimney?
Could you build a small batch box rmh in the basement? Or in a living room upstairs? Or even in a bedroom on the top floor?  
A small batch box rmh with a brick bell would not have a very big footprint and would go a long way towards helping to heat your home.
Batch box rmh's burn larger wood laid horizontal. No open flames visible. (Unless you give it a glass door) No chance of sparks popping out. Much hotter burn as well.
They also burn unattended, unlike a traditional J tube rmh that requires wood added every 30-40 minutes.

Just some information for you to ponder.  
Let us know if you want to build a rmh in your home, we will be happy to assist you any way we can!

I don't see why you would want to to be honest. thermal mass is designed to provide low level heating after the fire has burnt out, so you don't need to have a fire burning all night and therefore have to get up all night to feed it. Your oil furnace burns all night anyway without any input from you. Adding mass will not help it's heating ability, you may capture some heat that is going out the chimney but from my memory the exhaust on an oil furnace isn't all that hot at all.

We have underfloor heating in our bathroom, that works as a thermal battery but all it means is that if we do let the furnace burn out (we have both an oil and a pellet furnace on the same system) it just uses a ton more fuel to get the whole thing up to temperature again, there is no saving and in our case there is almost certainly extra losses from the floor.

Every type of burner needs an exhaust system that fits the design. I wouldn't want to modify a commercial oil burner until I could take the risk of it not working.
Adding a mass bench and routing the output side to of the oil burner (hot air?) would be sure to work, would be comfy, though probably not gain any heating efficiency. But for learning, you could learn to build a mass bench before taking on a custom stove project.