My solution has multiple start-up projects in it. When publishing to production, the projects start up, but then begin to act in unexpected ways. I'm guessing I don't understand the system well enough yet.
What happens to a web solution with multiple start-up projects when it is published to the web-server?
"Startup project" simply means which projects start when you press F5 in the Visual Studio IDE.
They have no impact on the deployed web application.
In the spirit of the holidays, I decided to ask myself if there was any way you could be correct in any degree other than zero. I found a "5%" possibility:
Are these other projects necessary for the correct behavior of your web application? You mentioned that they are "Console apps/scheduled tasks". Does your application require that these console apps or scheduled tasks are running on the web server before your web application will function?
If that's the case, then you should be informed that startup projects "have no impact on the deployed web application". The fact that you have this code listed as startup projects will not cause the console apps or scheduled tasks to run on, or even to be deployed to, the web server. You will have to deploy these .exe files to the server and you will have to start them yourself. "Startup projects" won't do this for you.
If that's not it, then I think this question should be closed as "not a real question" until you narrow down what your problem is. My answer has told you what startup projects do not do, but in order to solve your problem, you'll need to figure out what the actual problem is.