I'm not an expert, but it might lose it's tight roll shape if you iron it... but don't let the iron get too hot. Try the lowest setting and be gentle. It might even help to spray it lightly with water from a mist sprayer, then apply the iron. That will generate some steam which could also assist in getting the curl out of the grass mat roll.
The other possibility is to just unfurl it and lay it on a large patio table in direct sun. You might need to put some rocks on the corners to hold it down at first. The heat from the sun should help to release the curl tension from being rolled up. Just don't do it on a cloudy day or in an area where there are lots of birds that might let loose some droppings.
Hopefully, some of the experts on this forum can chime in with other suggestions.
Good luck!
Terrence
Long Island, NY