Your wheels are 16"x6", and Ford installed the biggest tire that will fit on those rims, per the tire manufacturers.
You can go up one smidgen with no problems, other than the tire makers all say you need 6.5" wide or wider rims. That size is LT255/85R16. I've seen the Cooper Discoverer S/T LT255/85R16 on a buddy's dually, they looked great, and he had no tire problems with them on his truck. Not many tire makers make that size, but there is a decent choice:
BFGoodrich MudTerrain
Cooper Discoverer S/T
Toyo Open Country M/T
Those 255/85s are 4.58 percent taller and a bit wider than your stock tires. So your speedo/odo/tripmeter will be 4.58 percent slower than it is now. If your speedo is dead on accurate now, with those 255/85s an indicated 70 MPH will be over 73 MPH per Officer Bob's radar. And an indicated ticket limit of 74 MPH will be a "Howdy, Judge" 77 MPH.
Some folks say your speedo is fast now and the 255/85s will made it close to accurate. Ask your GPS. Or Officer Bob.
Any Ford dealer can calibrate your speedo to make it more accurate. Most will charge about one hour labor rate.
Wider tires than 255/85s can be mounted on your narrow rims, but that can cause all sorts of problems of rubbing between the duals and rubbing on the front end. I wouldn't go bigger than LT255/85R16 on an '04 dually.