Yes. The "I Love New York" (I ❤ NY) logo is a registered trademark owned by the New York State Department of Economic Development (through its tourism arm, I LOVE NY). It was designed by Milton Glaser in 1977 pro bono for a state marketing campaign.
Unauthorized commercial use (merchandise, advertising, logos) is trademark infringement. The state actively licenses and enforces the mark. Non-commercial, personal use (e.g., a tattoo, a homemade sign) is generally tolerated, but any public/commercial exploitation requires a license.