The serial keys themselves are just text. Like this:
A039D-3HGWO-E297W-270CH-297CH
That text won't do you any harm, although serial keys can be blacklisted, even if the key works in the software, you might not get any support you might be entitled to if the key is blacklisted.
The real risks however come from the websites which host serial keys. They can contain malicious code and advertisements which are designed simply to exploit your browser. They often list fake keys or just list software names to get you to visit the site and infect you.
Some serial keys can come packaged in archive files. Remember that .ZIP and .RAR files can be harmful although the risk is much less than it used to be, self extracting executables (.EXE which extracts files) should be thrown away without ever executing them. As someone who was once many years ago in the reverse engineering scene, I can tell you serial keys are never packaged in .EXE files.
Then there are so called keygens, these again carry risks as even programs from legitimate cracking groups or reverse engineering groups can be bundled with malicious software and often are. I would recommend fully against using these.
Be wary of ignoring warnings from your anti-virus too. One common way to patch software is to inject into it's process and the code for doing this is nearly always flagged by antivirus as something like ProcInject etc. However, this is also a technique often used by malware and viruses to cover it's tracks. I mean just consider how much you want to trust these websites that you don't know. They're asking you to disable your antivirus and trust them. I wouldn't trust them
Hope that clears it up