When I look at our customer facing User Profile layout, I don't see any Influence thing that I can remove from the layout. When I look at what the customer sees, I see this influence box. Any idea on how to remove it?
First of all, this is not related to Campaign Influence which you have tagged at your question.
Secondly, it is good to have showing influence rather than hiding. It will then defect the purpose of using chatter profile.
For more information, refer Chatter Activity and Influence
Salesforce doesn't provide an option to hide this.
can't currently hide, here's the idea: https://success.salesforce.com/ideaView?id=08730000000I4uo