When a DC circuit is carrying current, large amounts or small, is there induced-emf due to the inductance? Or is it only applied to AC circuits?
In the limit of long times, the currents are steady, so the magnetic fields they create are steady so there is no induced EMF. This situation is usually tagged "steady state".
That said, there will be a period of time when you have just switched a circuit on or off during which things have not settled down and then there will in general be effects not seen in the steady state (including induced EMFs). This is called the "transient" behavior.
Transient behavior analysis is a important component of electronics design.
Here the use of DC in your question is a bad choice. DC does not imply constant current, it means that the current have same polarity in a period that we refer to. So you would like to use steady state in your question.
That said, in steady state the the current is not changing with respect to time so the flux is not changing and hence there is no emf. But if the current is not steady then there I will be an induced magnetic field produced which changes as the flux changes due to varying current. Ultimately this changing magnetic produces a non-conservative electrical field.