Why are some parts of the wings not protected by the wing anti-ice system in the B737?
There are always two answers to any question about why something is or is not on any particular aircraft: one regards regulation and the other practical application.
FAR § 25.1419, Ice protection, describes the certification requirement to allow transport category flight into known icing. No specific parts of the aircraft are listed, only that

(a) An analysis must be performed to establish that the ice protection for the various components of the airplane is adequate, taking into account the various airplane operational configurations; and


(b) To verify the ice protection analysis, to check for icing anomalies, and to demonstrate that the ice protection system and its components are effective, the airplane or its components must be flight tested in the various operational configurations, in measured natural atmospheric icing conditions and, as found necessary, by one or more of the following means...

The means listed and their explanation in various Advisory Circulars are rigorous, so the absence of icing protection for the leading edges of the 737 wing tips and roots is certified as adequate. This may seem obvious but regulations have been known to specify a particular solution to a problem, and could possibly have allowed certification without expensive testing if, for example, at least the middle 50% of the wing had a particular type of anti-ice installed.
Since the regs aren't responsible for this particular solution, we look to practical application. Bleed air anti ice is heavy, expensive, and reduces engine power output. Aircraft don't carry anything they don't need, so the question becomes: if you need a whole wing for an aircraft to fly, why do you only need to protect half a wing from ice accretion, and why pick the center section?
The answer to the first part of this new question has to do with critical phase of flight, which in this case is takeoff. The aircraft is heavy, slow and may need every bit of wing and available engine power. Diverting bypass air to heat the wing at this point is doubly punitive. The reason it is not necessary is because the wing (and control surfaces) are assumed to have been deiced/anti-iced by equipment on the ground. Beginning 1981 bleed air could be used to heat the slats during ground operations in the 737, and was automatically deactivated if engines approached takeoff power.
§ 121.629 Operation in icing conditions

(b) No person may take off an aircraft when frost, ice, or snow is adhering to the wings, control surfaces, propellers, engine inlets, or other critical surfaces of the aircraft or when the takeoff would not be in compliance with paragraph (c) of this section.

To satisfy this reg without ground de-ice you would have to heat two thirds of the airplane surface area, including the entire wing, which is wholly impractical. Assumption of ground deice/anti-ice allows design of onboard anti-ice to assume excess lift is available and you no longer need the whole wing. In the case of the 737, you only need to protect half the wing.
So why the center 50%? It's not always just the center, and engine power is (believed to be) the reason.

The NG series outboard slat has no wing anti-ice facility (see photo) believed to be due to excessive bleed requirements. However in June 2005 it was announced that the 737-MMA will have raked wingtips with anti-ice along the full span. This is because the MMA will be spending long periods of time on patrol at low level where it will be exposed to icing conditions.


The 737-MMA (Mult-imission Maritime Aircraft) is a modified commercial 737 airframe, known as the P8 Poseidon and described as

“A bit of JSTARS (Joint Surveillance Acquisition Radar System), a little bit of AWACS and a little bit of MC2A (Multirole Command and Control), but with the added ability to go and kill a submarine.”



Another unique feature of the P-8A is the raked wingtips. These, and the horizontal and vertical stabilisers, are electrically de-iced by electro-mechanical expulsion de-icing systems (EMEDS). EMEDS shakes the ice off the surfaces by using actuators in the cavity behind the leading edge. They can dislodge ice thicker than 0.15cm (0.06in).