I will try to be a little more clear, please forgive my poor english.
Your site is using sessions. Session are temporary files created by the site in a directory, and read back by the site to work properly. In this case, your site is using sessions to keep user data (i.e. username, shopping cart items, etc). Every site that works with sessions, needs to know where to save this temporary files, that are named always with the prefix sess_ plus a random string (i.e. sess_7eq8blaus5bfv85f4cqm1o9nn5 in your log file).
By default, every hosting server has a pre-defined directory where to save this files (most of them)... in a default ISPConfig 3 hosting server, this place is the /tmp folder inside the client folder. In your case, sess_ files should be created in: "/var/www/clients/client1/web1/tmp".
You can change this in realtime if you want to save sess_ files in another directory, php allows you to do it with the variable: session.save_path. This is what I think is happening here. In some place, your site is forced to use a different path to save sess_ files, as you could see in the log file, your site is trying to save them in /var/lib/php/session directory. As your site has no rights to access that directory, the file is not being created. When your site try to read it, the file doesn't exist, so, it logs an error in the log file.
With the "find" functions I posted, I was trying to guide you to identify where in your site this path is being changed by your site, without success at this moment. If you can find where this value is being changed, you can modify it to point to the right directory.
Are you using the site as fast-cgi? I hope this help you to solve your problem. Again, sorry for my bad english. I think that this is what is happening to you, I am not an expert.