No, at least not directly.
Unlike downloading a key generator or a cracked version of software, simply copy-pasting a product key will not directly have an impact on your system.
However, if the product key is tied to some sort of online account, it may be possible that an attacker can use that as an attack vector.
For example, imagine a software with some sort of vulnerability - for example a buffer overflow. An attacker would now somehow have to place user input into the vulnerable functionality. If the software retrieves values from an online account, this might be a sensible attack vector. An attacker would create an account, place the payload in the account, distribute the product key, and wait until others use it.
Another example may be that an online account + product key is used to create backups of the state of the application. You may thus accidentally expose sensitive data to the owner of the key.
I ignored the legal and ethical implications of the question, as they are not relevant to this site. But depending on your location, what you are doing is very likely illegal.