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.
That question has no clear answer (depending on transmission technology, energy-to-bandwidth ratio, compression technology/efficiency, compressibility of data, etc), leading to speculation - and making it off-topic here.
Also, high-volume content is most often compressed already - JPEG, PNG, MP3, and so on. Video codecs can have compression ratios beyond 500:1 - an uncompressed 4k video would require close to 5 Gbit/s. Trying to transparently compress these even further is utterly futile.
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.)