Can a state be removed from the United States without an amendment or would they have to choose to leave?
There is no mechanism described in the Constitution which allows for a State to secede. However the matter has arisen once or twice in the history of the USA (!) and the first principle, established both by the supreme court and on the battlefield is that States have no right of self-determination. A state cannot decide to leave by itself, whether that decision is made by the Governor, Legislature or in a referendum.
The Articles of confederation (that preceded the Constitution) make it clear that the founders intended the USA to be a "perpetual union". There is at least an argument, based on this, that nothing can allow a State to leave, neither law, nor war, nor Constitutional amendment!
But the Supreme Court, in analysing whether Texas had actually left the Union during the civil war, does say that the admission of a state is perpetual and there is "no place for reconsideration or revocation, except through revolution or through consent of the States."
This seems to be the highest authority: There is no Constitutional arrangement for a State to become independent so the only way for it to occur is the same way that the original 13 States became independent of Britain: By revolution, followed by either that state defeating the others in the war that follows the revolution, or by the other states consenting.
It is not clear exactly how the other states could express their consent. A convention is possible, which could draft an amendment allowing secession. But generally, the States can show their unified consent by acts of congress. It is a reasonable argument that a Joint Resolution of Congress, is sufficient to recognise the independence of a State.
The spirit of the Supreme court decision is that the admission as a State cannot be undone. It doesn't deal directly with the situation of expulsion. But note the term used: "consent". The states have to consent to the secession. This implies that the State must first demand to leave. It cannot be expelled against its will.