Is a tankless water heater more or less efficient than an electric stove at heating water? Consider the following scenario that I actually encounter frequently:  I just finished washing a pot in the sink, and now I want to boil some water for cooking noodles.  If I fill the pot with hot water from the pipes, my stove will require less energy to bring the water to boiling, but my tankless water heater will kick in and heat an equivalent volume of water to displace the water taken from the pipes.  Or, I can fill the pot with cold water, which won't trip the water heater, but will instead require more energy from the stove to heat.
It seems to me that if my goal is to minimize energy spent on heating my water, I need to know which is more efficient at heating water - the tankless water heater, or the stove.  I have seen this question, but it is about heating water on the stove vs. in the microwave, which is a different situation.
Which is the more efficient solution?
 A: I have thought about this too, and my conclusing is the tankless hot water is more efficient.
I have looked at how the flame in the furnace works and how the hot gasses are coupled to the water it is heating.  In mine at least (some model of Wyle McClain), the water flows thru two windings of copper pipe around a hollow core.  The copper pipe that is wrapped up has radiator fins around it individually for the whole length of the coil.  In other words, it's not just coiled copper pipe, but really finned coiled copper pipe.
The oil flame is injected into the hollow center of this double coil, where it directly heats in the inner layer of the coil assembly.  The hot gasses are then forced around one end to pass closely over the outside layer of the coil assembly.  The manufacturer claims a fairly high efficiency for the overall furnace unit, something close to 90% if I remember right.  I think that means that 90% of the energy of the flame goes into heating water, and only 10% goes up the chimney.
Now contrast that to a flame sitting under a pot with no fins at all.  Put your hand just over the edge of the pot and you can feel a lot of heat going up the side of the pot into the air.  That has got to be a lot less efficient at coupling the flame heat to the contents of the pot than the very carefully designed boiler optimized for that purpose.
There is one gotcha in this, which is that if the sink where you fill the pot is far enough from the furnace and hot water hasn't been used in a while, then a significant volume of hot watemr is wasted in the pipes.  If you have just used the hot water (just finished washing dishes as you said), or are about to use it, then this one-time per use waste will occur anyway, and I think it's a no-brainer that having the furnace heat the water is more efficient.
I have no real proof of any of this, but the argument seems rather compelling to me. You may be more convinced if you took out the hot water heating coil from your furnace and looked how carefuly everything was designed to maximally couple the flame heat to the water.  I was quite impressed when I saw mine.
