If the space elevator platform (The very top) is orbiting a planet in a geosynchronous orbit, wouldn't the forces on the elevator "rope" be minimal?
Supplemental to the other answers; you are correct that the net force on the tether would be minimal, since the rotation of the counterweight would counteract the force of gravity. But, the individual components of this net force aren't being distributed evenly.
Consider, say, the first kilometer of tether from the ground. This is being pulled down by gravity and "held up" by the counterweight, but the counterweight is many many kilometers farther away whereas gravity happens right there. So, the rest of the tether is being "pulled apart" by these two forces.
(Incidentally, the counterweight at the top of the tether is going to be well above what "geosynchronous orbit" would be for an ordinary satellite. Instead the center of gravity for the entire counterweight-tether system would be at geosynchronous orbit altitude, or a bit beyond.)
Imagine building a stack of dirt. For a short period of time, it might stick straight up, but eventually it will fall down, forming something of a pyramid. Every material has an amount at which you can build with it and not have it collapse. Going to a pyramidal shape helps somewhat, but it doesn't solve everything. Specifically what is important is the Breaking Length, which can be defined as the maximum weight of an equal area vertical structure that can be supported for the given material. If a material can sustain a 5000 km vertical tower (As measured at sea level), it is strong enough to use for building a space elevator.
A space elavtor works by balancing gravity with centrifugal force.
As you get further from the center of the earth (which is both the center fo gravity and the center of rotation) gravity decreases and centrifugal force increases.
Each segment of the "rope" can be assigned an "effective weight" consisting of weight minus centrifugal force (both weight and centrifugal force are proportional to mass). For segments below geostatoinary orbit the effective weight will be positive, for segments above geostationary orbit the effective weight will be negative.
Tension in the rope is highest at geostationary orbit where all the rope with negative effective weight is pulling on all the rope with postive effective weight.
While the effective weight of the rope below geostationary orbit will be lower than it's actual weight it will still be a substantial fraction of it.
We can to some extent work arround this by making the rope non-uniform but just as with a rocket you need high ISP to avoid ludicrous mass ratios with a space elevator cable you need high strength to weight ratio to avoid ludicrous thickness ratios.
The cable is under tension - maximum in the center - because every foot you go up the cable, that's one more foot of cable hanging from it.
This can be addressed by tapering the cable: the cable cross-section increases as you approach the center in order to support the increasing amount of cable hanging from it. There is a 'taper ratio' equation (I read an article about all this, IIRC published in either "New Destinies" or "Analog" at least ten years ago). Given carbon fiber, the taper ratios start getting down to (again, IIRC) about 4.5.