Putting on blast means exposing someone. When someone says they are going to put someone on blast, it means they will publicly expose them, criticize their actions, and share their personal information with others.
It usually has a negative connotation as it involves sharing damaging and embarrassing details about the person to a broad audience through social media or any other platforms.
Example
One of the famous examples of putting someone on blast is the controversy between Stranger Things star Noah Schnapp and Doja Cat, where Noah exposed his chat with Doja in which she was asking to be set up with his co-star.

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