Currently there are Honda Ridgelines, Hyundai Santa Cruz, and Ford Mavericks that actually have front wheel drive. I think Honda Ridgeline is the first one to have a front wheel drive midsize truck. I would consider them as  midsize pickup trucks, but with a different purpose in mind. As for full size pick up trucks, I haven't heard of them being front wheel drive.

As @legendaryslayer says, the Maverick, Santa Cruz and Ridgeline are probably the closest things you can find to a truck with FWD. I wouldn't call them "trucks", myself, because of how they're powered and built compared to an actual, body-on-frame pickup truck. I would consider them to he crossovers. 

Putting any kind of weight in the bed of a truck pulls the center-of-gravity closer to the rear axle. Filling a bed with gravel bags, sand, or other heavy things causes the bed of the vehicle to be pushed slightly downwards, which also slightly lifts the truck on its front wheels. For a RWD truck, like an F-150, this is advantageous, as the additional weight increases the traction and pushing power the rear wheels can provide before the tires slip (think of people who put snow in their truck beds). The front tires lose a little bit of their capacity to maneuver the vehicle compared to when it's empty, but you're usually going slower, so this doesn't matter so much. This configuration will pull things uphill quite well, because the weight shifts to the rear, driven wheels.

For the unibody, FWD "trucks" like a Ridgeline, putting weight in their beds has the same effect on the suspension, only they are slightly lifting weight from the drive wheels. This causes a loss in power transfer capacity and turning capacity with the load. For the use they're made for, basically an SUV with the roof and 3rd row removed, made for light duty hauling, they work fine. In more testing situations, like pulling loads uphill, you lose even more pulling power because of the same effects, only they are all working against you. For a suburbanite who commutes and gets a few plants, etc from the hardware store, this is a perfectly fine configuration. It's not good for real truck work.

Just for a extra info related to the discussion, forklifts place their drive wheels at the front and turning wheels at the rear to maximize their traction when they're carrying heavy objects. This increases the traction on the drive wheels, while slightly reducing the steering capacity of the rear wheels. You don't care about this with a forklift, as they drive very slowly. They will roll if you turn too quickly. 

Front drive pickups are nothing new.