Yes teeth whitening gel does expire eventually and should not be used after the expiration date.
The teeth whitening gel expires around 12 months after you purchase it.
The teeth whitening gel should have expiration dates on them however some may not so if you've had the teeth whitening gel for longer than 12 months after purchasing it then it's not really good to use.
The teeth whitening gel loses it's effectiveness after it expires so it's best to use some fresh teeth whitening gel if you want to get the desired results.
Teeth whitening gel also oxidizes overtime and when it has expired it will change color because of the oxidization process.
So throw the expired teeth whitening gel out and get some fresh teeth whitening gel.