I don't think it does because cookies are only generally used for the essential functioning of the site, for example signing in to gitea, making comments etc.
It has web analytics but all data collected is anonymised and none of it sent to third parties like google.

You can see what data the analytics collects here.

All the data and reports get deleted every day.

You can also see how the apps are built here.

I would turn on "do not track" in chrome chrome://settings/privacy.