Well, no ... you dont' have to sign a new lease while the old one is still in effect.
As to how much notice is required, there is probably a clause in your lease that will control notice for rent increases. If not, the landlord probably doesn't need to give more than 7 days notice that the new lease will contain a rent increase. (If its a month to month lease, the landlord needs to give 30 days notice.)
If you're in a rent-controlled district, of course, local laws proabably give you additional rights.
Remember, though, that rentals are pretty freely available these days so you may avoid the rent increase by simply finding a cheaper place.