The basic assumption written into all of the code is: "there is a tile". If you use meshes, EVERY SINGLE PIECE OF CODE THAT REFERENCES A TILE MUST BE CHANGED.
That is... almost everything, as Putnam said. It's a good suggestion, but a lot harder than you think it is.
It's a lot of code to be changed, but it's very often just about how far away something is affecting the distance in which an interaction occurs, or something equally simple.
Combat? The range in which you can fight with any particular part becomes the distance of the used part from the head (which becomes measurable with the tags I proposed).
Sound? Goes from being a large cube to a simple distance.
Chains? Again, from cube to distance.
A dragon aiming fire? Goes from aiming at the tile a creature is in to aiming straight at the creature (adventurer mode would still use tiles though)
Rooms? Still uses tiles.
Minecarts colliding with walls and ejecting their contents 1 z-level up? Change that to giving the contents a small vertical velocity.
Minecarts deciding they've collided with a wall at enough speed to cause an eject? Check to see if the cart lost more than 5m/s of speed.
Occasionally you get something that isn't trivial like dances, but that's still not that difficult.
Lots of them can even still reference tiles if needs be, until mechanical stress is a thing mechanisms will always be dead-centre on a tile.
A lot of things have to be changed, but the changes are almost always extremely simple tweaks to the simpler things in the game, the bulk of them could probably be run through in a few days (the toughest of them, mainly one's I've mentioned previously, would take much longer though).