What happens if an aircraft goes above maximum altitude?
There is a variety of reasons for a maximum altitude:


Most common is insufficient engine performance to climb any higher. Then your question never arises. The aircraft simply cannot go higher.
For more powerful aircraft the partial pressure of the oxygen supply in an unpressurized cabin would drop below the value to keep the pilot(s) alert. For GA aircraft without supplemental oxygen this would be 15,000 ft (short term) or 12,500 ft (long term).
For many airliners it is the pressure limit of the cabin which is certified for a maximum differential to outside pressure. If the airplane climbs higher, the internal pressure must be reduced to less than what is deemed the acceptable minimum for the passengers. 
For special subsonic high altitude aircraft with sufficient engine power the limit is the coffin corner when minimum speed and maximum Mach fall together. When that height is exceeded, the aircraft stalls and exceeds its Mach limit at the same time. The result is a shallow dive which can only be recovered a few thousand feet lower.
In some cases the limit is given by minimum combustion chamber pressure inside the engine to keep the flame burning. There are cases when aircraft climbed too high, suffered a flame-out and could not restart.
With supersonic aircraft the limit might be set by maximum flight Mach number, when higher speed means stagnation heat exceeds structural limits. The high Mach number is needed to sustain minimum dynamic pressure to produce enough lift.