Does dizziness from metoprolol ever go away?

Not if it's caused by metoprolol causing too slow a heart rate or too low a blood pressure. To avoid lightheadedness, keep your awake resting heart rate no lower than 60/minute and blood pressure no lower than 100 systolic.

As Dr. Colantino correctly states, this is a blood pressure/ pulse issue which is characteristic of this drug class (beta blocker). If the effect/ side effect mix is too challenging, suggest switching to a different pharmacological class. A second opinion may assist in this endeavor.