How to close the current extension tab?

Most likely you're running your code from a content script, where chrome.tabs is undefined. If this is the case, you can instead send a message to the background page and have the background page (which has access to chrome.tabs) make the call.

Note that from a background page, you would use chrome.tabs.getSelected since getCurrent will return undefined.

It works for me with one little fix:

chrome.tabs.getCurrent(function(tab) {
    chrome.tabs.remove(tab.id, function() { });
});

Just make sure you're really running this code in options page of your extension and not just some HTML page, because chrome.tabs API is available only for extensions.