I think you could. Cyanotype stands up to dry mounting, so you could probably get away with an iron low setting, or put it in a press at about 180f. Of course any abrasion is just that, so a measure is a measure. Just don't wash it in any detergent that has phosphates!
You can.
I printed cyanotypes on silk and ironed them, and not only once. I even used the steam (otherwise I would not get rid of the crinkles), but did take care to iron only the backside and to use a neutral baper base on the ironing table.
The problem with cyanotypes on silk is rather how to get them anywhere near strait in a frame, or on a canvas frame for that matter.