The major issue with landing a plane that is overweight is over-running the runway.
If you are trying to land a plane that is over it's rated landing weight, you will need to be going faster to keep enough airflow over the wings to generate the lift required to keep the plane in the air. This, of course, has the consequence that it will take longer to stop when on the runway.
Another factor that contributes to it taking longer to stop when on the runway is that the extra weight of the plane adds to the momentum. The calculation for momentum is p=mv where m is mass and v is velocity (for this example we can think of it as speed.)
So not only are we now landing with a higher mass (m) we are also landing with a higher speed (v) so our momentum increases very quickly.
For example, if we had a plane that was 660,000 pounds, and needed to travel at (lets say) 290km/h (156 knots) to keep giving enough lift, it's momentum would be:
191400000 pounds km/h
If we look at a normal landing weight and speed of a 777 524,000 pounds landing at a normal speed of 250km/h then the momentum would be:
135500000 pounds km/h
you can see that even a small increase in speed causes the momentum to increase significantly.
The other issue is the load put on the landing gear which have the potential damage them, although I think that the deceleration speed is the bigger factor here.
(I am not a mathematician, my calculations may be wrong so feel free to edit the answer if I did get them wrong)
EDIT: One other thing I forgot to mention was that most larger airports have a special kind of surface at the end of the runway that serve to slow planes. They are constructed (iirc) out of some sort of very strong foam like material that is very effective at slowing a plane. I don't remember exactly what it was made out of though.
As mentioned by NoviceInDisguise and ChrisW, this is actually called the Engineered materials arrestor system and is made out of lightweight, crushable concrete blocks.
You can see an image of the effect here where the plane's wheels have dug through the soft layer which helps slow the plane.: