How does Newton's Third Law of Motion affect a roller coaster?

The Third Law is often summarized in the aphorism, "for every action there is an equal and opposite reaction." In more precise terms relating to physics, this means that when a force acts on an object, it will always experience an equal force directed back at itself due to electromagnetic repulsion.

This makes a little more sense if we establish some terms. Typically, at least when considering everyday examples of physics, we're looking at objects with mass interacting with other objects with mass. Force can be defined as mass accelerating, which means it's changing its speed or direction. This is accomplished by bringing the masses in contact with each other, so the clouds of electrons surrounding their atoms begin to interact and repulse. The more they are forced together, the greater the repulsion. Since the electrons and their atoms don't "know" who's doing the pushing, as far as they're concerned, there's no reason for one of them to experience all of the force, and the other to experience little or none. The force is an interaction rather than an "energy dump."

In a roller coaster, this can be best seen when the roller coaster changes speed or direction; for example, you feel heavier at the bottom of a drop because you're being pushed toward the seat, but the seat is pushing back. Part of the reason for this is that you're actually experiencing a slight delay; your body is still moving down and "catching up" with the seat even after it's stopped moving down. This is also why you feel weightless during the drop; the seat has basically fallen out from under you. Instead, the seatbelt or safety bar is pushing you downwards so that you fall relatively at the same rate as the car. 

Overall, the main effect of the Third Law is that the roller coaster car, and the frame of the coaster itself, experience considerable amounts of stress during the ride. Every time the roller coaster car accelerates, force is applied both to the car and to the frame, which is part of why coasters require constant maintenance to ensure they're safe.