No, your premiums will not go up if you make an uninsured (or underinsured) motorist claim. If you were not at fault for the accident, your auto insurance carrier cannot raise your rates.
After your auto insurance carrier settles your uninsured motorist claim, they will likely go after the uninsured driver and attempt to recoup the money they paid out to you.
Because there are so many uninsured (and underinsured drivers on the road) we strongly recommend that all drivers protect themselves by carrying uninsured/underinsured motorist coverage of at least $100,000 per person, $300,000 per accident. Better still: $250,000 per person, $500,000 per accident.