Will ovens emit carbon monoxide when off at high altitude if not properly adjusted?
High altitude conversion kits for gas stoves are an actual thing you can buy. The kit decreases the amount of natural gas sent to the burners, because of the decreased quantity of oxygen available at altitude. Here's an installation manual that mentions that not installing the kit where needed can result in incomplete combustion. This will result in CO output.

Now, the news video isn't playing correctly for me, but it sounds like they were saying levels were around 22 PPM. Normally, this is not yet at the level to cause symptoms, which starts around 70 PPM, though it's obviously still worth tracking down the source. However, the high altitude plays a role here as well. This document states specifically that "at high elevations the onset of symptoms occurs at much lower concentrations of CO than expected". Is 22 PPM life-threatening at that altitude? Probably not, (lethal range starts around 400 PPM at normal elevation) but I wouldn't want to stick around to find out.

In the end, if you have gas appliances, make sure you have a CO detector, and make sure it works.

TLDR: They are talking about a real phenomenon, but I doubt this is a life-or-death issue.