For fun (and for future evil diabolical schemes) would humans have the ability if given the finances and resources, to use some kind of massive wrenching system to pull different continents together? What kind of effect would this have on the world? Would it be possible to shift the continents? What kind of material would be strong enough to withstand that kind of force?
No. It's not possible.
The forces involved in the movement of continental plates are absolutely immense. even if you could work out a way to apply that amount of force the strength of the material would not support it. Push on one part of the continent and you would create a localized mountain range "upstream" and a valley "downstream" but the rest would just keep flowing on around you.
You would need to somehow apply the incredibly enormous amount of force you require spread out over the whole body of the continental plate down to the depth of miles. There is no technology we currently know of or could conceive of that would allow us to do this.
The most amazing and interesting thing is that the continents are already in motion! Naturally, that is. This is known as plate tectonics. Basically there are two processes at work here:
1- Deep in the sea at some points (known as mid-ocean ridges), the surface of seafloor spreads sideways and slowly the layer beneath comes up. This is known as Seafloor Spreading.
2- In other places, the crust is slowly pushed downwards and into the mantle. These are known as subduction zones.
The combined effect of both these processes is that the crust swims on the molten mantle and continents, along with the crust, move about. This movement is very very slow however and a continent only happens to move a centimeter or two in a whole year.
The force for this immense motion comes from the heat energy of the mantle. The forces involved are so huge that they cause earthquakes of magnitude 8 and greater on the richter scale. This involves energies which would be emitted as a result of several megaton nuclear bombs explosions!
Theoretically if you apply enough force in the right direction, at the right times, you might be able to alter this process to suit your directions. The stress is on the word theoretically. It would be practically impossible to alter or stop this process with our current technology, no matter how hard you try and how much time/money you invest for it.
In short: No.
The issue with humans moving continents is that it assumes that the continents are being moved by forces acting on them laterally. It's not - what moves the continents is acting on them from underneath, through the mechanisms of mantle convection and gravity.
And gravity is the main one. Up to 95% of the force moving a subducting slab consists of slab pull as the subducting portion of the slab is pulled down into the mantle.