California Vehicle Code, division 3, chapter 1, article 1, section 4000:
A person shall not drive, move, or leave standing upon a highway, or in an offstreet public parking facility, any motor vehicle, trailer, semitrailer, pole or pipe dolly, or logging dolly, unless it is registered and the appropriate fees have been paid under this code or registered under the permanent trailer identification program, except that an off-highway motor vehicle which displays an identification plate or device issued by the department pursuant to Section 38010 may be driven, moved, or left standing in an offstreet public parking facility without being registered or paying registration fees.
(Section 38010 defines "off-highway vehicles", essentially agricultural, construction, and other vehicles that are never driven on state roads)
Article 2, section 4156:
(a) Notwithstanding any other provision of this code, and except as provided in subdivision (b), the department in its discretion may issue a temporary permit to operate a vehicle when a payment of fees has been accepted in an amount to be determined by, and paid to the department, by the owner or other person in lawful possession of the vehicle. The permit shall be subject to the terms and conditions, and shall be valid for the period of time, that the department shall deem appropriate under the circumstances.
Article 7, section 4850:
The department, upon registering a vehicle, shall issue to the owner two partially or fully reflectorized license plates or devices for a motor vehicle, other than a motorcycle, and one partially or fully reflectorized license plate or device for all other vehicles required to be registered under this code. The plates or devices shall identify the vehicles for which they are issued for the period of their validity.
Article 9, section 5202:
A license plate issued by this state or any other jurisdiction within or without the United States shall be attached upon receipt and remain attached during the period of its validity to the vehicle for which it is issued while being operated within this state or during the time the vehicle is being held for sale in this state, or until the time that a vehicle with special or identification plates is no longer entitled to those plates; and a person shall not operate, and an owner shall not knowingly permit to be operated, upon any highway, a vehicle unless the license plate is so attached. A special permit issued in lieu of plates shall be attached and displayed on the vehicle for which the permit was issued during the period of the permit’s validity.
Divsion 17, article 1, chapter 1, section 40000.1:
Except as otherwise provided in this article, it is unlawful and constitutes an infraction for any person to violate, or fail to comply with any provision of this code, or any local ordinance adopted pursuant to this code.
In short, yes, you need a license plate or equivalent temporary registration permit to drive on the highways; California is a bit unusual in that it requires license plates for parked vehicles as well.