I wonder if a network, for example internet, will take more or less energy if more compression is enabled. More compression would take more CPU and thus more energy but on the other hand it would be less data to transmit.
It's a common misconception that it takes less power to not transmit user data. The truth is, every interface that's "up/up" is continuously transmitting -- either user data, or an idle pattern. It's not like the laser is off when there's no packet to send. Without digging into the vagaries of signal encoding, it's a fairly uniform power draw.
When you add in compression, you're adding additional hardware and/or increasing the processing (computational) load. That hardware and processing takes power. How much depends on a number of factors: efficiency of the hardware, complexity of the compression algorithm, interface speed, etc. The reason to use compression is not to save power, but to save bandwidth. (i.e. make a slow link a bit faster.)