Will the tip of my nose drop after the cast is removed? (Photo)

It is very early to suggest the tip was turned too high.  From the front, you appear to have adequate tip rotation and not too excessive.  Wait til the cast is removed and 6 weeks after that to start considering these issues.  Looks good so far! 

It’s not unusual to see changes in the nose in the immediate post-operative period.  This occurs for a variety of reasons including the healing process and the resolution of swelling.
            When surgeons perform rhinoplasty they utilize a series of detailed maneuvers on the underlying bone and cartilage.  In the initial post-operative period, these changes are often obscured by inflammation and swelling.  In some cases, as swelling resolves and nasal contour improves, the nasal tip may actually appear to drop.  The nasal tip should be more or less stable after the resolution 

Appears as you missed the pre operative discussion concerning swelling and healing following rhinoplasty. PLEASE return to your chosen surgeon to fully understand the time of swelling and healing. You as SOOOOOO early in the healing phase that very hard to help you understand via the internet... 

The tip often drops a bit after surgery. Right now it is taped up and maximally swollen. Best to take a deep breath and give it time to heal.