Does heat affect the magnetic field of an electromagnet?
No, heat has no influence on the strength of a magnetic field produced by current flowing around a coil of wire.  The strength of that magnetic field is strictly the result of the ampere-turns of current going around.

However, heat can effect the magnetic permeability of various materials.  If the electromagnet has anything other than a air core, then how the field resulting from the ampere-turns is concentrated and shaped can differ with temperature.  This concentration and channeling of the magnetic field can make a electromagnet appear to have a stronger field, and can make it act "stronger" in many applications.

For example, let's say you wrap 100 turns of wire around a wooden rod and put 1 A thru it.  The magnetic field strength is strictly a function of the 100 ampere-turns of current going around.  However, this magnet will be able to pick up heavier objects if the wooden rod is replaced by a iron rod of the same shape and size.  This is because the iron is a much better conductor of magnetism than air and wood are, so the magnetic field lines will be concentrated at the ends of the iron rod.  This more concentrated field is able to pick up heavier magnetic objects as a result of this concentration, even though the overall magnetic field has the same average strength in both cases.

In the example above, the apparent strength of the electromagnet with a iron core depends on material properties of the iron, which can vary with temperature.  The magnetic permeability of free space is not effected by temperature, so the same coil without a core would make a magnet that does not vary with temperature.

Of course extreme tempertures change the wire and will eventually melt it so that you don't have a electromagnet anymore at all.  That obviously changes things, but I'm assuming that's not the kind of effect you are asking about.