how would a modify headers add-on be able to let the server think that you are coming from a different address? It's not as if the browser sends the origin address as a HTTP header.

It doesn't have to. It's part of the IP packet which contains the TCP packet which contains the request headers.

Are you sure that you are not using a proxy server at the address Wolfram gives you?

Edit: On second thought, you could try to fool server-side detection by setting a non-standard X-Forwarded-For header, but a what is my ip service shouldn't trust that and just report the real remote address.

Actually W|A always have the result available in alt tag of the image:

    alt="IPv4 | 124.120.*.*\nIPv6  (short version) | ::ffff:*:*\n(as seen by Wolfram|Alpha)" title="IPv4 | 124.120.*.*..."

So Sed still works ;)