For reference, let's take the A320. The nose is painted like the rest of the body.
Areas on the airplane with ice protection have it for important reasons:
Ice on the wing/empennage disrupts airflow, adding drag and making it more susceptible to stalling.
Ice on an engine cowl lip can disrupt flow into the engine, causing stability issues. If it dislodges it can damage the engine.
Ice in the pitot/static system prevents the air data system from reading airspeed or altitude properly.
Ice on the windshield can make it harder to see through.
Ice in drain lines can cause them to clog and not drain properly.
The nose is not as important. Ice will add drag, and has some risk of breaking off and damaging something. But this is not as critical as with other areas. There is also an added challenge in adding anti-ice measures, as it must interfere as little as possible with the weather radar inside. It's made from composite material, which doesn't conduct or cope with heat very well, so using either bleed air or electric heating would be an issue.
Wings need undisturbed airflow in order to create lift. Even a small covering of ice can lead to a significant loss of lift. Noses do not create lift, all ice accumulation may do is increase the weight of the airplane a small amount, which isn't a big enough concern to warrant the cost and weight of an ice prevention system.