What part of the government would manage this? President? Congress?

I would imagine the Congress.

Now that we have citizenship, there is a whole new world of possibilities...

Already suggested -> Judicial System

Great idea, Would limit chances of PTO's if you know who is a PTO candidate

Congress can manage deportation like how they manage granting citizenship.

Or an office can be appointed by President to manage it?

If it were permanent, then there needs to be a way to overturn it... such as a future administration overturning the previous ruling.

How would that work?

This could be abused, instead of buying ticket you can ask someone in congress to transfer you out of the country :)

This can work like the Citizenship that is now implemented... Anyone can be placed on a list and then it is voted on by the Congress like a bill. If it passes everyone on the list is kicked...

BUT where are they kicked to???

In the last state where they have been

States should definitely be able to remove citizenship and deny users (or even whole nations) from entering into or working in their country, just to make it more realistic.