No it doesn't, if you:

1. Never register with OpenAI/ChatGPT, and

2. Strongly block ads, e.g. using uBlock Origin + EFF Privacy Badger. Yes, those don't work on Chrome, Edge and other Chromium-based browsers.

Also, even then - ChatGPT may be tracking your behavior indirectly through Microsoft's various services and platforms. But we should do our best to undermine mass surveillance and support individual privacy.