PA-28 flight instructor reported a stuck throttle due to a carburetor failure resulted in an immediate landing.

Narrative

I was on a training flight with a CFI applicant; we departed ZZZ at XA:30 local time; and traveled down the coast towards ZZZ1. After several touch and go's; we departed the area and proceeded northeast; to the local practice area off ZZZ. The student demonstrated slow flight and a power off stall. After the completion of the power off stall; I instructed the student to demonstrate a power on stall; as he set up for the maneuver; he told me the throttle was stuck in the current position that it was in. (Roughly 2300 RPM.) The throttle could travel forward to maximum or back to the position it was being stuck in; but not lower then that power setting. After trying to troubleshoot by adjusting the throttle friction and to overpower the throttle being stuck; it was determined that it could not be fixed in air. I then contacted Approach [Control] and requested radar vectors to ZZZ. I informed Approach that we would have to pull the mixture once over the runway at ZZZ in order to land. We flew the plane to ZZZ; and landed without incident or any damage to the airplane; person; or property around XC:00. Upon getting the plane back to our maintenance shop it was determined that our airplane sustained a carburetor failure. The accelerator pump fuel needle dislodged and blocked the butterfly valve inside the carburetor throat. This in return; blocked throttle movement.