Since this question came up today, I figure it was worth sharing in case others are working in the Federal space or an environment that has strict requirements that no data is being transmitted from their environment and need to remove the Feedback Icon (Smiley Face) in the vSphere Client. In this post, I will assume that you're running vCenter Server Appliance 6.7 or 6.5 Update 2.
Update (8/20/2018): These steps are applicable for vCenter Server 6.5 Update 2 and later. The vSphere Client build must be 126.96.36.19900 or later.
Note: I want to preface the following workflow as Not Supported by VMware at the moment, and should only be performed in a lab setting.
Lets start off by accessing the vCenter Server Appliance via SSH, jumping into the BASH shell (shell) and navigating to /etc/vmware/vsphere-ui, the directory in which the configuration files for the vSphere Client reside.
Using VI or any other text editor, edit the webclient.properties and insert the property feedbackTool.enabled. By default, feedbackTool.enabled is set to true and not exposed in the properties file. We'll be overriding that with a value of false, which will force the vSphere Client to load within this enabled.
Insert the following.
Note: You don't need the comment, but its always nice to tag any modification you've made.
#Flag that toggles the Feedback Tool Smiley Face
Once complete, go ahead and restart the vSphere Client service (vsphere-ui) via service-control, and you'll have disabled the lovely little Smiley-face-of-Feedback from all of the users that log into the client!
It should be noted that if you have multiple vCenter Server in (Enhanced) Linked Mode, this process will need to be repeated for each of the vCenter Servers as this setting does not propagate.