@Casey is correct. Nothing will happen under normal operating conditions. The FAR §25.831 Ventilation requires,
... the ventilation system must be designed to provide each occupant with an airflow containing at least 0.55 pounds of fresh air per minute.
In general, the amount of air available is far greater than the requirement. On the ground, compressed air for the Environmental Control System, which supplies air to crew and passengers can be obtained from an APU, a GCU, airport high-pressure hydrants, or the aircraft engines. In flight, however, compressed air is obtained almost exclusively from the compressor stages of the aircraft engines.
The pressure would not be affected by this. According to FAR §25.843, Tests for pressurized cabins, the pressurized cabins are tested to show the performance of
... the pressure supply, pressure and flow regulators, indicators, and warning signals, in steady and stepped climbs and descents at rates corresponding to the maximum attainable within the operating limitations of the airplane, up to the maximum altitude for which certification is requested.
Note that in some aircraft the air conditioning system is switched off during climb- this is to do with the load on engine and not the air supply.