Can I route coaxial cable for tv through a gas fireplace?
DO NOT DO IT!!!!

Chimney vents must not be compromised! All heat and gas related stacks must be left intact. No additional venting or other holes may be put in them. As the comments have noted, you run the serious risk of carbon monoxide poisoning, as well as the risk of chimney and house fires.

Find another route well away from the chimney.