First up: I am not a lawyer.
Being German and knowing the little legal nitbitts of the laws related to our social security system, I can tell you that it won't have direct legal consequences.
BUT: If you get sick or get into an accident, it will get very expensive… especially if you need things like surgery.
My personal advice is to apply for unemployment benefits (ALG 1 or ALG 2, depending on how long you've worked here etc.). Please note that German law dictates that you have to apply for such unemployment benefits as soon as you know your contract ends… and that up to 3 months in advance.
So, because you already know that your contract will end, simply apply for ALG 1 (or 2) right now! To avoid waiting lines, you should know you can do all that via regular mail.
Even if there's a chance you won't get cash from them for only a few days, at least you'll have complete health insurance. After all - you never know. Bad things can happen, and you surely don't want to be financially ruined if they happen.
Better be safe than sorry… it won't cost you more than a few minutes and some postage stamps to apply.
Hint/Tip: law also dictates that a "formloser Antrag auf Arbeitslosengeld" can not be rejected by the "Arbeitsamt" or "Jobcenter". Due to that, they will send you the correct forms for you to fill out and that's about it. Not long after you've returned the filled and signed documents, they will invite you for a talk to assess your individual situation... and depending on the details, they'll guide you through the rest of the procedure. But don't worry, filling and signing the forms is about the hardest of it all. ;)
EDIT (a note aside)
For your convenvience, I've let the “Bundesagentur für Arbeit” send me their current brochure, which - as you'll see - goes hand-in-hand with the clause in your contract about registering for unemployment benefits when the contract ends… in fact, as you already know it'll end, be aware of their "3 months in advance" clause.
Sadly enough, they don't have an English version of that brochure… at least, they stated they don't have an English one when I called them up. Nevertheless, as a "side-note" edit, this should satisfy.