I doubt it very much. There are accounts in the medical literature of rashes and other cutaneous symptoms arising during iron therapy but even in these cases, the dosage was far higher than found in multivitamin preparations. I can't find any recorded instance of iron making hayfever worse.
In my opinion, it's far more likely that you may have an allergy or hypersensitivity to another substance contained in the multivitamin preparation. Don't forget that you not only need to look at the therapeutic ingredients listed on the back of the box or container but also the excipients (the pharmacologically inert substances used to bind the ingredients of the multivitamin tablet together) such as various emulsifiers, lactose, hydroxypropylmethylcellulose etc. These substances and others that might be in the tablet are known to cause allergies.
I appreciate that discovering exactly what you're allergic to in these tablets is a daunting task so I've got a few suggestions to make.
Firstly, try taking the multivitamin preparation every three days for a fortnight and see if it still makes the hayfever worse. Secondly, change the iron-containing preparation for one that is iron free - many branded multivitamin preparations nowadays come in both versions. Thirdly, try taking an alternative brand of multivitamin tablets, with or without iron and see if that makes a difference.
These steps should give you an idea if it is truly iron that's making your hayfever worse.