I have an iOS app that is send a POST request to my server and then it runs a function and returns some data back to the app.
My question is if my app has lets say 100 people using it at the same time and make the POST request in quick succession, is this going to case an error? 
I am on a shared server if that matters.
No it should not cause an error, you are requesting the server with an HTTP method(POST) which serves every request from a new user as a new request because HTTP is a stateless protocol it doesn't save user states(unless you are managing user sessions explicitly)
So if 100 users are requesting at the same time, they are different, each request is different and same will be the response(different) for each user.
And as others said, its completely dependent on the server how it handles the request.
Send multiple POST requests a once