What are the legal implications of not signing a summons?

You don't sign a summons when served with one; you simply have to have it handed to you, whereby you are recorded as being the recipient of the summons. It is the issuer who has to sign it. If it's not signed, then it would be legally invalid.

However, after being served with a summons, not abiding by its terms will most likely result in a bench warrant for your arrest from the issuing judge. And then the agency issuing the summons can send out arresting officers and you go to the local lockup (police station), followed by a trip to the area local jail to await your trial. (Here in Chicago, Cook county, this is 14 days.)