Yes, in electrical circuits of only passive elements (Resistors, Capacitors, and Inductors) only the Resistors dissipate power (as heat). Active elements like transistors can also dissipate power, and if the currents in the circuit are changing with time, then power can be radiated away in electromagnetic waves. Therefore, the electrical resistance of the magnet coil dissipates power.
Additionally, power may be dissipated in the iron core of the electromagnet or in iron the magnet is picking up. As the DC current is turned on, a voltage is induced in the magnet's iron core by the changing magnetic flux. This causes power to be dissipated in the resistance of the iron. Likewise, as the magnet is brought near a piece of iron being picked up, the magnetic flux in the piece changes, a voltage is generated, and power is dissipated by the piece's resistance.