Is the electric field zero inside an ideal conductor carrying a current? By an ideal conductor, I mean one with zero resistance. Inside an ideal conductor with no current, the electric field is zero, but is the electric field still zero with the ideal conductor carrying a current?
 A: If the conductor has zero resistance, then the force required in order to sustain the current is zero, (the electrons keep moving by themselves) so yes, the electric field is zero.
As real example of that, an electric current flowing through a loop of superconducting wire can persist indefinitely with no power source !!!
A: NOTE: This answer relates to original question, where ideal conductor was not specified to be the one with zero resistance.
No.  The electric field is the requirement for the electrical current.  This can be seen from the Ohm's law in th vector version
$$\vec{j} = \sigma \vec{E},$$
where $\vec{j}$ is current density and $\sigma$ is conductivity.  Hence, no electric field, no current.
A: 
but "ideal conductor" usually means that the conductor conforms to Ohm's law.
I am surprised. I always heard " ideal " as " zero resistance when constant current flows ", hence zero potential drop along the wire. Hence the answer to the original question is yes, by definition.
Conductor that conforms to Ohm's law is called " ohmic ".
A: Yes, it is zero.
In short
Ideal conductor, means no change in electric potential along it, which means no electric field inside.
In details
1) Assume there is a non-ideal conductor of length L.
2) Assume there is some non-zero current flowing through it causing potential drop across the ends of the conductor.
3) Electric field inside a conductor is uniform and parallel to the surface of the conductor.   
$$\Delta V = \int {\vec Edl = EL} $$
$$\Delta V = IR$$
Now if one assumes conductivity tends to infinity (resistance tends to zero), electric field tends to zero.
$$E = {{IR} \over L} \to 0,{\rm{ if \: }}R  \: \to 0$$
