Long story short = they'll lose an enormous amount in regards to revenue due to it.
The people who dislike the comments the most are the majority of the authors of the articles. They hate the bickering and they constantly complain to editors that the comments are pointless and draw away from the real story that they slaved so hard to write.. They're happy the comments are gone. Some would even demand comments be turned off on their articles.. which they allowed to be turned off from time to time.
But this affects revenue in a serious way. The adverts they post on their page are directly related to hit counts. At first CNN will try to say their hit counts are more accurate now and each hit is worth more than before since there's not a plethora of hits from one user refreshing the page... But the number of visitors will decline rapidly also. So not only are individual hits, but also unique visitors will both go down. Once the unique visitors drop, the cost advertisers pay CNN to show their adverts will drop substantially.
All this will happen because many 'readers' at CNN are there for the comments and comments alone. They read the headline, and then straight to the comments. Others are there to read a article and then express their opinion about it. And of course some are there to just read articles and they pay little to no attention to the comments. Once the comments are completely (or almost completely) shut down, a large portion will stop frequenting CNN as much. And once their traffic diminishes, their revenue from adverts diminishes, and then they'll bring the comments back.
You, hell even THEY might think that they're doing something good...they're keeping people from arguing, and making it easier on themselves...but what it all boils down to ...is the main one calling shots cares about revenue first and foremost... give it 2 to 3 months and comments will be back in full force. All because they want more money.