They make headphones that use this technology, I don't think it could be easily applied to a PC case. Quieter components/passive cooling/sound deadening materials are simple & easy to implement. Active noise cancellation requires a great amount of precision, otherwise you'll end up worsening the situation. The two sound waves must reach the ear at exactly the same point in time, for instance. Example, if you were sitting between the device & a person talking, or a speaker playing music, etc. you would hear the devices attempts to cancel the noise a fraction of a second after you heard the noise itself.
Headphones work because they act as a doorkeeper, all sound going to your ear must pass through them (for all practical purposes, anyway).
Viper GTS