If you can't set up caching rules on Apache (as suggested by @David Dorvard's answer - see the part using the <Files
directive), you could pipe the image through a PHP script, and set your own (no)caching headers there:
<?php
header('Cache-Control: no-cache');
header('Expires: 0');
header('Content-Type: image/jpeg'); // or whatever your image is
readfile('/some/path/to/yourfile.jpg');
?>
This should get you an image that's not cached at all; emphasis on should, as various browsers are variously broken (IIRC IE6 cached it anyway when linked as background-image, but thankfully that's on the way out).
Note that this simplistic approach will 1) increase the load on the server as it needs to start PHP for the image request and 2) disable partial download on the script/image