Hmm. Sorry I'm not all that familiar with Flickr, but the PHP error is reasonably understandable.
It's trying to load up the API.php file from the Flickr/ directory. It'll search your entire PHP include_path to find it.
Now, your provider has control over your include_path. Usually it contains the current directory, listed as ".". To find our what your include_path is, just create this little PHP file on your webserver and run it:
The output will tell you a lot about your PHP installation, including the include_path. The directories will be separated by colons. PHP (and therefore s9y) will look for the file in each of those directories.
So, what's "."? Usually it's the directory of the currently-executing PHP file. According to your warning, that's /var/www/s9y-lab/plugins/serendipity_plugin_flickr/serendipity_plugin_flickr.php, so "." is /var/www/s9y-lab/plugins/serendipity_plugin_flickr/. Is there a /var/www/s9y-lab/plugins/serendipity_plugin_flickr/Flickr/API.php? The message implies that there isn't.
So, after writing all this, I find the plugin at look at the README. It says you need to follow the directions at
http://code.iamcal.com/php/flickr/readme.htm to get the thing to work. That's about installing the PEAR Flickr API. PEAR is to PHP what SPARTACUS is to S9Y, essentially. Most providers include PEAR, but most don't provide the Flickr API. Yours certainly doesn't.
Instead, you'll want to download the package listed in the INSTALLATION section and unzip it into your plugins/serendipity_plugin_flickr/ directory. Then things should start working.