I recently found (here: Does every web request send the browser cookies?) that every HTTP request contains the cookies related to a domain every time a request is made to that s
You can strip cookies with a proxy server. For example our product WinGate will allow you to modify requests (and responses), and you could use this to clear the Cookie header in requests.
However, this will prevent a large number of websites from functioning properly, as cookies are used to transport session IDs so that the server can identify each connection / request your browser makes as being from the same "session". HTTP itself does not have any concept of session.
Disclaimer: I work for Qbik who make WinGate.