Discounting the extra rolling friction (tire) which takes relatively VERY little extra fuel to overcome, and assuming we're talking about a flatbed trailer with little wind resistance, the only extra energy required is to accelerate the trailer up to speed and pull it up hills.
Now if you talking about a livestock, box trailer or camper trailer, then overcoming the extra air resistance takes the same extra fuel no matter what the trailer weighs and is only a function of your speed and additional surface area. Depending on the profile of the trailer, this can add a LOT of air resistance that needs to be overcome with extra fuel.
Imagine this: 2 trailers, same profile, same rolling resistance, but one weighs twice the other. If you were to drive 300 miles continuously on flat ground at the same speed with both, the only fuel difference would be what you used in the 1st 1/8 mile getting it up to speed.