So I just installed pinhead locking skewers on my bike, using the instruction manual. But the wheels now have a large amount of friction. I installed all the pieces as tightly as possible, could that be the cause? Or do I need to grease the skewer first?
The skewer isn't an axle so nothing rotates and it and there's no need for grease. It's more likely that the wheels aren't completely straight in the dropouts causing the brakes or tyres to rub. I tend to tighten the skewers with the bike's weight on the wheels, rather than on a work stand. But you should also check the sequence of components again as well.
Finally the tab washer supplied with pinhead skewers these days may be the issue. The single tab goes in the dropout, after the skewer. It's just possible that even installed correctly this washer doesn't sit in the right place and misaligns the wheel. One of my bikes had pinhead skewers from new 7 years ago without these washers, with no problems; I've just added them (when fitting matching skewers to my new bike).