How does electricity produce heat, and where do the electrons go?

No, current is not lost due to heat. The same current flows in one end of a wire that flows out the other end, regardless of how much heat is dissipated in the wire (or any other resistor).

Energy is conserved because the resistance times the current causes a voltage drop. This voltage times current is power, which is taken from the circuit and heats the resistor. The voltage drop in the resistor means there is less voltage available to the rest of the circuit, but the current thru the whole loop is the same.

This is similar to a turbine being driven by high pressure water. The same amount of water comes out of the turbine as goes in, but there is a pressure difference between the input flow and the output flow. It is that pressure difference (analgous to voltage) times the water flow (analogous to current) that represents work done, which goes to spinning the turbine.

The electrons don't "go" anywhere, and current (the net flow of electrons) is not "lost" to heat. But the electrons gain energy through the application of an electric field, and they can lose that energy through inelastic interactions with the other particles (nuclei) in the conductor. The energy lost is in the form of randomized vibratory motion, which is just another way of saying "heat".

If you want a more detailed answer, you should ask on the Physics SE.