I am not sure how much resources the combat log uses, but here is something I did to try to limit the resources used by the log in your UI. I don't think you can turn combat logging "off" per se, since your client needs to get a lot of that information for other reasons, and because AddOns can always request combat log events. But what you can do is apply a minimal filter to the log.
Create a minimal combat log filter:
- Right click on the Combat Log Tab and Choose Settings.
- Click 'Add Filter' and give it a name.
- Select the filter, and in the panes below, de-select all the types of events.
Once you do this, you'll have a new filter for the log (in addition to 'Everything,' 'What happened to me?,' etc). When you use this filter, you'll see no combat events appear in the log.
I am not sure what kind of performance impact this has. I believe it does save resources, since combat events won't be formatted for display. However the events are still sent to your computer. So I'm not sure, but you can give it a try!