Pour some liquid bleach on the dead animal. Then scoop it up with a shovel or pick it up using plastic bags for gloves and put the dead carcass in a garbage bag. Tie the ends of the garbage bag tightly, so smell can't escape. To be on the safe side, put the garbage bagged animal in another garbage bag and tie it up properly, and in order to get rid of the odor, gasoline can be used instead of liquid bleach outdoors. Read more on this from; http://www.ehow.com/how_4503022_rid-dead-animal-smells-minutes.html#ixzz1fRiKsleG.
Now you know how to keep the rat without it being smelly... :-) In order to keep your dryer as well, you could try to put a bowl with baking powder in and wait some days. Baking powder absorbs smells. Or may be a anti-smell spray, but I?m not sure how effective those are. But for a proper job you should take the thing apart and clean the affected parts with water and detergent or disinfectant.
like david o said,you should get rid of the animal the way he said to. but instead of using gasoline to get rid of the smell, i would use lysol wipes or clorox to clean up the germs and diseases. if u do this over and over each time it dries(5 times at the most), you should then be safe to continue drying your clothes.