I don't really get why it SHOULD move at all...
First of all the magnetisation of the iron seems irrelevant here.
If you powered a straight coil of wire with an conductive (iron) core inside, the increasing magnetic field would induce a current
in the core.(An "eddy current" to be exact)
However, the Lorentz-force acting on that current would (under the approximation that the Magnetic field inside the coil is perfectly straight) be either inwards or outawards.So there would be no net force on the core and it wouldn't move.
You can check that by using the left hand rule.
I don't see any reason why it would be different for a toroidal solenoid.
But there IS a phenomenon where inductive currents do make conductors move.
If there is a divergence or convergence in the part of the magnetic field where the conductor is,(like the ends of the coil) and you increase the magnetic
Field, then it will be shot away from the source of the magnetic field.
Here is a video of it
Sadly they don't actually talk about WHY the ring flies away at all.And i won't either since that wasn't your question.
But i am confident that if you are interested in it you will find it out with Lenz's Law and the left hand rule.(maybe a diagram will help ;) )