The Live HTTP Headers extension for Firefox can be a useful debugging tool, as it allows Web developers to see the sent and received HTTP headers for any page. There is no such extension for Opera, but you are not without options. The following are some possibilities:
In Opera 10, Tools - Advanced - Developer tools -> Network. This is a complete solution.
In Opera 9.2, Tools - Advanced - Developer console -> HTTP. This works in exactly the same way as a custom button. See that section for further details.
9.2 version is the same as a custom button, except it also works with pages in framesets. The 10 version has no known limitations.
A bookmarklet or custom button can run a script that uses XMLHttpRequest to request the page again, and see what headers are received. This is the most limited of all the approaches.
Bookmarklet: HEADers (right click and bookmark it).
Custom button: HEADers (drag it to a toolbar).
Some local proxy servers can display sent and received headers. Of these, by far the most popular is Proxomitron, and it is the one I personally recommend, if you are able to use it for your project. As well as showing headers, it can optionally be used to add, remove, or modify content on the fly. Proxomitron is Windows-only, but can be run on a different computer to the browser itself if you have access to a Windows computer, and in theory it also works under wine.
To use it, install the proxy server on your computer. Start it, disable all filters, and open a log window, where the sent and received headers will be displayed. In Opera, use Tools - Preferences - Advanced - Network - Proxy servers, and set HTTP to use localhost on port 8080 as a proxy server (the default port used by Proxomitron). You can then quickly toggle it on or off using Tools - Quick preferences (or F12) - Enable proxy servers.
Another example would be Fiddler (Windows only), which seems to default to using port 8888.
This is basically the same approach as the Live HTTP Headers extension for Firefox uses (except that it transparently reads the headers as they are sent and received, and does not need to use an external local proxy), and is the best approach for most developers.
Network packet sniffers operate on the network itself and watch the HTTP communications. Some of these can be difficult to setup and use, as they are often designed to do much more than just HTTP. The most popular is Ethereal, but there may also be some specialised tools just for HTTP which could be easier to understand.