After performing additional research on this issue, I noticed that "legitimate" requests to this script include a "src" query string parameter, e.g., the string "?src=/path/to/requested/image.jpg". If I specify the path in that format, the script returns the expected output (an actual thumbnail image).
So, I'm beginning to wonder if these warnings aren't simply the result of search engines or bots that are obtaining invalid URLs from one place or another, and not so much the hallmark of a true "problem".
In Apache's access.log:
/var/www/example.com/log/20130609-access.log:199.21.99.115 - - [09/Jun/2013:05:23:13 -0700] "GET /wp-content/themes/ElegantEstate/timthumb.php?src=http://example.com/wp-content/uploads/2012/09/TestImage-1. HTTP/1.1" 400 214 "-" "Mozilla/5.0 (compatible; YandexBot/3.0; +http://yandex.com/bots)"
And the corresponding entry in Apache's error.log:
/var/www/example.com/log/error.log:[Sun Jun 09 05:23:13 2013] [warn] [client 199.21.99.115] mod_fcgid: stderr: PHP Warning: file_exists(): open_basedir restriction in effect. File(/var/www/example.com/web/wp-content/themes/ElegantEstate/timthumb.php/wp-content/uploads/2012/09/TestImage-1.) is not within the allowed path(s): (/var/www/clients/client7/web23/web:/var/www/clients/client7/web23/tmp:/var/www/example.com/web:/var/www/example.com/tmp:/srv/www/example.com/web:/usr/share/php5:/tmp:/usr/share/phpmyadmin:/etc/phpmyadmin:/var/lib/phpmyadmin:/usr/share/squirrelmail:/etc/squirrelmail:/etc/mailname:/etc/hostname:/usr/share/php:/etc/roundcube/:/usr/share/roundcube:/var/log/roundcube:/var/lib/roundcube:/var/www/clients/client7/web23/protected:/var/www/example.com/protected) in /var/www/clients/client7/web23/web/wp-content/themes/ElegantEstate/timthumb.php on line 849
I noticed that the HTTP GET request from access.log includes the entire URL (including protocol and host). So, I tried including the full URL to a real/existing image, and that works as expected, too.
I then noticed that the file name of the requested image is truncated in the GET request: the real file name is "TestImage-1-Large.jpg" (and the GET request includes only "TestImage-1").
It seems the bottom line is that TimThumb is simply poorly-written and doesn't perform a few simple checks that could eliminate this junk from my error logs.
In any case, I don't think this is worth troubleshooting any further. I really appreciate the feedback nonetheless.
Cheers!