Diagnosis
Yeah, the screen has a standard matte anti-glare coating that can get gummed up. Using any cleaner with alcohol, ammonia, or abrasives will permanently damage that coating, making the smears permanent.
How to Fix It
Unplug the TV. Use a dry, clean microfiber cloth first to lift off dust. For fingerprints and smudges, slightly dampen a different microfiber cloth with distilled water only. Wring it out completely so it's just barely damp. Wipe the screen gently in a circular motion. Immediately follow with a dry section of a clean microfiber cloth to buff it completely dry. Never spray liquid directly on the screen.
If That Didn't Work
For stubborn spots, you can use a 50/50 mix of distilled white vinegar and distilled water. Same method: dampen the cloth, never spray. The mild acidity can cut through grease. Buff dry immediately. This is still safer than most commercial cleaners.
Heads Up
Never use paper towels, window cleaner, kitchen spray, or compressed air (which can blow dust into the seams and speakers). Pressure can damage the panel. If you see rainbow-colored smears after cleaning, you've damaged the coating. That's not fixable.