Not really.
Off load there are two principle losses, magnetizing current and hysterisis loss.
The magnetizing currentis, as as been said, at 90 degrees to the voltage but it is still real current, and will result in a current loss, reactive power. It is quite definitely not loss free.
The hysterisis loss is the energy consumed sweeping the magnetic flux around its BH curve. This also is not loss free.
A small transformer, 50VA, shouldn't consume more than 10% of its rated load in these losses. If it does then it is a poorly designed and built transformer. A toroidal transformer will be lower, it is more efficient.
Just leave a transformer connected but off load, after some time you can feel that it is warm, may be reactive current but still losses.
On no load the power factor will be close to zero, reactive losses, and will improve as the load comes on and real power is transmitted through the transformer.
Almost any book published prior to 1960 will explain all of this. Anything newer seems to be full of differential equations, which unless you can weild them are not very obvious.