How can one check to see if a remote file exists using PHP?

You can instruct curl to use the HTTP HEAD method via CURLOPT_NOBODY.

More or less

$ch = curl_init("http://www.example.com/favicon.ico");

curl_setopt($ch, CURLOPT_NOBODY, true);
curl_exec($ch);
$retcode = curl_getinfo($ch, CURLINFO_HTTP_CODE);
// $retcode >= 400 -> not found, $retcode = 200, found.
curl_close($ch);

Anyway, you only save the cost of the HTTP transfer, not the TCP connection establishment and closing. And being favicons small, you might not see much improvement.

Caching the result locally seems a good idea if it turns out to be too slow. HEAD checks the time of the file, and returns it in the headers. You can do like browsers and get the CURLINFO_FILETIME of the icon. In your cache you can store the URL => [ favicon, timestamp ]. You can then compare the timestamp and reload the favicon.