Screw the cap back on a plastic water bottle while in cruise, by the time you land the bottle will be crushed due to the change of cabin pressure.
If overhead bins were airtight, they would have been forced open or ripped during a climb. Best case scenario they will fail (e.g. crack) slowly after repeated cycles.
Answer is no, they are not.
Related: At which point does an airliner start to pressurize/depressurize the cabin? A typical cabin altitude is between 6,000 and 8,000 feet during cruise for a jetliner.
While this is best suited for Health.SE, the smaller the mammal the faster the breathing / heart rate. And if panicked, it is even more. In a confined space the CO2 from breathing won't be replaced quickly, which can eventually lead to asphyxiation.
See this from Biology.SE: Sitting in a closed room filled with regular air - for how long will the air last?
Although, overhead lockers are not air-tight, lack of oxygen could have been the cause of the dog's death, US media report.