DC through an inductor
It might help considering all possible cases and then go down to the particular case you are interested in.
Much in the same way as current is the time derivative of electric charge, you should see voltage in a circuit with magnetic elements as the time derivative of magnetic flux.
The voltage you see at the terminals of an inductor is due to the time derivative of the magnetic flux linked by the inductor path... and the jump between terminals (or the rest of your circuit, if it does not enclose other variable flux regions). This is a consequence of Faraday's law.
Now, what can the cause of this changing magnetic flux be?

It could be a changing magnetic field due for example to a magnet moving relative to your stationary coil. In this case, you can have a voltage at the coil's (the inductor's) terminals even when they are not connected to a circuit.
If you close the circuit - for example through a resistor - that voltage will show up at the resistor's terminals and a (variable) current will flow in your circuit and in the inductor, producing an additional magnetic field. The additional changing magnetic flux linked by the inductor will oppose the original flux (according to Lenz's law) and is associated to the self-inductance L of the coil.

It could be a changing flux due to a current flowing in a nearby coil. In this case too, you can see a voltage at your coils open terminals. This voltage is characterized by the mutual inductance M.
If you close the circuit on a resistor as before, you will cause the flow of a changing current that will create a changing magnetic field. The additional changing magnetic flux linked by your inductor - characterized by the self inductance L - will alter the voltage you see (and will also affect the 'primary side' through the common mutual inductance).

It could be the changing flux due to a current flowing in the coil itself when it is driven by a generator. In this case the self inductance L of the coil is the only culprit for the voltage you see and the circuit must be closed in order to drive your coil. The changing current imposed by the generator will cause a changing magnetic field, and the changing magnetic flux linked by the coil itself will produce the voltage you see at its terminals.


Therefore, if you are in this last case, where the flux comes from the current in the circuit that contains the inductor, when you have no changing current (because DC means your circuit has been on since t=-infinity or long enough to reach equilibrium), you have no changing flux and hence no voltage at the inductor's terminals. The only voltage you will see is that due to the ohmic drop in the conductor your coil and your other circuit elements are made of.
(The really funny things with inductors, any inductor, is that the voltage becomes path dependent and while you can see the EMF in the jump at the terminals, if you follow the coil the value of the voltage will be zero)