That's a very thoughtful question. Yes, heating on part of a metal makes it easier for electrons to move away from it, although obviously very few can leave before electrostatic forces put a stop to that. So to get any appreciable current, you need a circuit. The question then becomes, which way will the electrons flow around the circuit? If it's all made of one type of wire, then the directions are equivalent and no current will circulate even if one part is heated.
In order to get a current to keep flowing, you need a circuit with two different types of material in it. Since the electrical push created by the temperature difference will be different in two different materials, the push from the hot-to-cold part of the circuit doesn't have to cancel the push from the cold-to-hot return.
This thermoelectric effect is used to make a type of thermometer and also, running in reverse, to make devices that pump heat from one region to another when driven by a current.
There's a nice Wikipedia article discussing these effects and their uses.http://en.wikipedia.org/wiki/Thermoelectric_effect
(published on 07/04/2011)