Does double brick chimney need liner for vented gas logs?
If you have a fire in that fireplace, you really need the chimney relined. Its not that it will be any deposits, that is true, but there will still be a possibility of carbon monoxide leaking into the home, if not other fumes.

There are companies that will clean the interior, place a long balloon inside, inflate it and pour a product around it to seal the brick.

Another way to do it is to have a semi-rigid stainless steel pipe set in place. I have only seen this done with furnaces, not fireplaces, but there may be a way nowadays to tie it into the smoke chamber, after it has been restored.