expect(...).toThrow() is for checking if an error is thrown from a function call. When calling an async function, it never throws an error; rather it returns a Promise which may eventually become “rejected.” Although async functions use the same throw/catch terminology, the code required to detect a thrown error differs from what’s required to detect a rejected Promise. This is why Jest needs a different assertion technique.
Try expect(...).rejects.toThrow() instead:
await expect(api(`verify/${profile.auth.verifyToken}`, {method: 'POST'}).json())
.rejects.toThrow();
Notice you have to await this assertion because Jest needs to wait until the Promise finalizes before seeing whether it resolved or rejected.