I have a commercial web application that I’m using. During a demo that will be attended by my competitors I want to hide the companies branding. I will likely be presenting from my surface. Could I use docker to substitute or mask the logo images and text that the app sends over?
Short answer: not at all 
You are talking about a web site, so HTML or CSS can achieve this, Docker can run it only.
You could use a HTTP proxy to manipulate the HTML code.