What are the general guidelines for determining when flux should or should not have rosin?
Rosin-free flux is acidic. If you don't remove the residue, it can cause corrosion. These are typically used on non-electrical solder connections.
Rosin flux is not acidic, and the residue will not corrode. Ordinary electronics solder contains rosin flux.
There's three general kinds of solders (not counting lead-free vs leaded).
1) Solid core solder: used for plumbing with acid paste flux you put on copper fittings with a brush. If you don't get the flux into the joint before you start soldering, the flux likely won't make it deep into the joint resulting in a joint that will prematurely fail.
2) Acid core solder: used for putting together sheet metal or other kinds of metals that you decide you don't want to braze or weld together. The acid core is corrosive which allows the oxide layer to get taken off resulting in an easier soldering job.
3) Rosin core solder: used for electronics. This solder contains flux within the solder that isn't acidic. If you used acid-core on electronics, the acid flux left afterwards would start etching your copper traces and soon you'd have open circuits all over your pcb.