Non-zero primary current under open-circuit test
Primary inductance

Primary current needs to remain finite to keep the core magnetised. It's the change in this magnetisation that generates the voltage in all the transformer windings, both primary and secondary. As the transformer becomes 'better', higher permeability core, smaller air-gaps in the magnetic circuit, this current becomes lower, and in the ideal transformer = infinite permeability/inductance case tends to zero. 

This current only stores magnetic energy, so is reactive, and is represented by a shunt primary inductor in a transformer model.

While it does dissipate some power in the primary winding due to \$I^2R\$, as the magnetising current is usually an order of magnitude or two below rated current, this loss is usually negligible.

Core losses

Hysteresis loss in the core is due to the finite amount of energy required to take the core magnetisation round the full BH loop every cycle. Transformers tend to use 'soft' cores that have narrow BH loops to minimise this loss.

Eddy current loss in the core is due to closed conducting paths around areas of changing flux in the core acting as short-circuited parasitic secondaries. Cores are made out of thin insulated laminations to reduce the area and hence voltage round these loops, and high resistance materials to minimise the resulting currents flowing.

Both of the core losses dissipate energy, so look resistive in the transformer model.