Are virus signatures a serious threat to the bitcoin network?
The pastebin you linked describes the harm caused as follows:


  What'll happen is that that will trigger the anti-virus program for
  every bitcoin client user that has an anti-virus program installed
  (and this won't go away ever.. you can't clean the database). Those
  virus signatures are harmless in their own, but they are something
  that anti-virus program use to recognize viruses. The anti-virus
  programs will think that the local bitcoin database is infected.


If a transaction manages to trigger an anti-virus alert for someone, I'd blame it on the anti-virus program being badly designed. 

Yes it will probably be annoying for a bunch of users but Bitcoin already has a strict enough rule set to make this hard to happen. 20 bytes isn't a lot, for most computers, a single standard word character is 1 byte. If a virus program goes around looking for suspicious data that is 20 bytes long they will run into a lot of false positives. And I doubt that there is any seriously dangerous virus that is under 20 bytes in size.

TL;DR

It's very hard to pull off anything harmful, you will need to develop a virus that only takes up &lt;20 bytes and can destroy a system. Otherwise anti-virus programs would show you a warning that you can possibly close/ignore, it's up to the anti-virus programs to be smart enough to discern.