This is a question we get asked all the time and the short answer is that you should maintain at least one server if you are concerned with the security of your network and users. The most important function a server provides, other than host databases, files and applications, is the capability to centrally manage the security in your network. Also, a server allows you to efficiently and remotely push software and updates to all PCs in the network (i.e. Java updates, Office updates, Printer updates, deployment of remote management software such as Logmein, etc.).
Essentially via a security policy implemented on your server you can control what each user on the network can or cannot do on their local computer. For instance, you can completely remove the control panel for certain users and their capability to install programs. You can also enforce specific firewall setting for each computer and reset individual passwords in a centralized manner as required.
Furthermore, you can extend your server security to integrate with your cloud services. This is typically referred to as Single Sign On(SSO). Basically, by signing in to your computer, SSO allows you to access all the apps you have access to without having to sign on to each individual cloud app.
In a nutshell, if you have moved all your apps and data to the cloud, you should eliminate any servers that used to host data and apps, and maintain a minimum of one server. This move reduces your IT management cost while maintaining your ability to centrally and efficiently manage the security of your network and users.