The “Streisand Effect” will certainly occur.
https://en.wikipedia.org/wiki/Streisand_effect
Although this seems like a modern, Internet-era phenomenon, variants of it have always existed. My grandfather did not drink very much before, during, or after Prohibition except maybe a bit around Christmas and New Years. However, during Prohibition, he made “bathtub gin” and gave it away. I have heard that it was surprisingly good.