The show's chief medical examiner, Dr. Ducky Mallard, examined Special Agent Tony DiNozzo and instructed him on strict voice rest for 24 hours which Tony ignored.
If I was on the show, I would have performed a much more thorough examination that would include fiberoptic laryngoscopy (and/or stroboscopy) in which the vocal cords could be directly examined as shown in this video:
Based on his hoarseness, lack of productive cough, and acute onset, I suspect Tony developed acute VIRAL laryngitis. Here's an example of what I suspect I would have seen on laryngoscopy if one was done (video is actually stroboscopy, a much more intensive examination than laryngoscopy):
This exam is different than acute BACTERIAL laryngitis in which pus on the voicebox would be seen as shown in this video (also stroboscopy):
Overall, treatment for acute viral laryngitis is strict voice rest as the esteemed Dr. Mallard recommended, though 24 hours is a bit optimistic. I probably would have prescribed a course of steroids to help with the swelling as well as hydration.
Should Tony keep on talking in setting of acute laryngitis, he runs the risk of developing muscle tension dysphonia.