What happens if you exceed the maximum speed on a C172?
You can, but you have to live with the consequences. There are several things that can happen:


Depending on the vertical gusts ahead, you might not even get close to v$_{NE}$. There is another speed limit for gusty weather called v$_B$, and exceeding this will run the risk of overstressing the wing structure. Going above v$_B$ will overstress the wings in a gust of more than 50 ft/s, and more than 25 ft/s when flying above v$_D$. The exact values can be found in the flight envelope diagram of the flight manual.
On a calm day, you can fly v$_{NE}$ and even a little faster, but once you fly fast enough, flutter will become very likely. Note that flutter needs some initial excitation, so you might fly well into the flutter speed range before flutter occurs. When it does, control surfaces will be ripped from their fittings which will make the aircraft pitch up. At that point the wings will break off.
The engine on a C 172 is not powerful enough to sustain level flight at v$_{NE}$. You need to dive the aircraft, which requires some altitude. Maintaining the speed will mean that you dive into the ground, so you need to pull out of the dive in time.
On really fast aircraft the pitch trim will get more nose-heavy when the aircraft approaches the speed of sound. You don't run that risk in a C 172, but faster aircraft found themselves locked into a dive which they could not end.