There are irons and steamers in the laundry rooms on the Dream....at least in the one on Deck 2.
Here is the list of prohibited items, as listed at the
DCL site:
Question
What items are not permitted on board?
Answer
In order to maintain a safe and secure environment for our Guests and crew, the following are examples of items that Guests are not allowed to bring on board.
These and other similar items will be confiscated upon being found.
Weapons and firearms (including ammunition, knivesincluding dive knivesswords, daggers or any other types of weapons) are not allowed onboard under any circumstances
Realistic replicas of firearms and weapons
Highly flammable liquids and combustible substances
Non prescribed controlled substances, illegal drugs, medically prescribed and synthetic marijuana
Power Tools and Tool boxes (except those tools of the trade utilized by vendors providing services)
Sporting equipment (i.e., baseball and cricket bats, hockey and lacrosse sticks, pool cues, ski poles, fishing gear and archery equipment.)
SCUBA equipment (i.e., tanks, spears guns /slings, dive knives, pry bar, and underwater
scooter)
Kites, balloons, surfboards, wave boards, inflatable floats/pools (including pool noodles), and wagons
Bicycles, Segways, skateboards, inline & roller skates, any footwear with wheels, (i.e. Heely's type shoes)
Electrical appliances such as rice cookers, hot plates, coffee makers, electric irons, bottle sterilizer and bottle warmers, etc. Please note: A limited number of approved bottle warmers and bottle sterilizers are available upon request through Guest Services. Also all ships offer laundry facilities with irons and ironing boards. Valet laundry service is also available for a nominal fee.
Candles & Incense
DVD, VCR, or Blu ray players or, Playstation, Xbox, and Nintendo Wii gaming systems are not allowed because of compatibility issues with the stateroom TVs
Musical instruments
Large Ice Coolers (Please note: small, personal-sized coolers, no larger than 12" x 12" x 12" brought onboard for the purpose of housing medications are permitted as carry-on luggage.)
Homemade, pre-cooked or other perishable food items plus any open snack containers
Metal detectors are not allowed on the ship or on Castaway Cay, as any object found using thesedevices would need to be returned to the owner or donated to charity
Personal fireworks or pyrotechnics
Seasonal and celebratory lighting strings
Liquid oxygen
The following items are permitted but with the following restrictions:
Walkie-talkies and two-way radios (e.g., Motorola, JVC). Please be aware that the amount of steel in the ship's infrastructure may cause some "dead zones" and the radios may not transmit in these areas.
A small portable fan. However, please contact Guest Services to have the Chief Electrician approve the use of your particular appliance in the stateroom.
Coast Guard approved personal flotation devices (PFDs) and flotation sewn into the bathing suit for safety. However, recreational floating devices such as floats, rafts and fun noodles cannot be brought on board and are not permitted in the pools or on Castaway Cay.
Golf clubs, as long as they are stored in your stateroom.
I bolded the the sentence as that may mean steamers as well...but it is not very clear. I would call DCL and ask them.
MJ
Mine is a small travel steamer; I think I got it at Target for about $30. IMO steamers are safer than curling irons or hot rollers, but if they say no then of course I wouldn't bring it again.
It's definitely worth the money to not have to stress out like that! I would pack all of my nice clothes separately and send them off as soon as you can...then just relax. 
