If I am applying for a job at a restaurant would it be unprofessional to eat at that same restaurant after picking up an application?
I would not only say that it's not unprofessional, but I would highly recommend it. It will give you a chance to observe things from a different perspective and watch how people are treated there as well as the general atmosphere (at least from a public-facing perspective).
It would also give you the opportunity to try the food because it's very difficult to be enthusiastic about a place if you don't believe in the product they're selling. Additionally, it could give you something to talk about if you have an interview. "I was here last week and I really enjoyed eating here and ..."
No, it would be a good thing. The only caveat is that you should act professionally while there as a customer. Simple stuff - be polite, leave an appropriate tip, etc.
Be observant while you are there. It gives you a better idea as to whether this is the kind of place you want to be part of. Generally, you can tell how a company treats their employees by how the employees treat the customers.
Also, sometimes the question comes up in the interview - Are you a customer? Now you can say "Yes".