I have a situation where getting additional servers will be impossible. Could I potentially host a provider-hosted app on the same server as SharePoint? What problems might I run into?
We have done this with multiple solutions without any problems.
Create a new web application for the provider hosted app. Do additional configurations (https://msdn.microsoft.com/en-us/library/office/fp179901.aspx) and you are good to go.
It is not recommended, but it can be done. I think there is a plethora of reasons why you don't want to do it: performance, security, etc. The whole point of it being a provider-hosted app is that it is provider-hosted, vs. SP hosted.
There isn't any reason it can't be on the same server, but best practices are to separate those provider-hosted apps from SharePoint.