When you use Slack, you don't "use it in a web browser", you use the (proprietary) software, delivered by Slack Technologies, which happens to run in a web browser. The fact that the web browser is itself free software is as irrelevant as the fact that the kernel and user-space services supporting skype on linux/bsd systems are open source. The actual software, everything that actually relates to the task at hand (group communication), is non-free.

Actually it's not completely irrelevant, as the fact that the host platform of proprietary software is free software has value in the security-based argument in favor of free software, but that's only tangential to the topic at hand.

Would it also be bad to host the project's source code on GitHub

Not necessarily, if really all you're using github for is hosting, then there's nothing wrong. That is, if you don't use their issues/PR system or other auxiliary services. This is how, for example, the linux project uses github. In this case, all that you're asking people to do is use git (free software) to download some stuff from servers owned by Github, Inc.

Running the website on IIS is fine though; this imposes absolutely nothing on your users or contributors.