DC current will build a static field in an inductor. If it is iron based core it will move the magnetic S curve of core flux density to one side eating up part of the available non-saturated core range.
For a use like a switching power supply flyback, buck, or boost converter there will be an average DC bias on the core. This must be taken into account on sizing the core so it doesn't get into saturation with the AC current riding on top of the DC average. When you start to saturate the core the inductance value drops.
For a case like a push-pull primary switching power supply (like found in most portable AC inverters), if each side switching device (MOSFET) does not balance out there can be a DC bias developed in the core limiting its power transferring capability. If this DC bias moves the core saturation curve enough to one side the AC primary currents will drive the core into saturation causing a current spike in primary side devices making them overheat or burn out.