Can a postsynaptic neuron 'shut itself off?
An example would be the neurons that switch off/switch on based on our sleep cycle. Nerve-signaling chemicals called neurotransmitters control whether we are asleep or awake by acting on different groups of nerve cells, or neurons, in the brain. Neurons in the brainstem, which connects the brain with the spinal cord, produce neurotransmitters such as serotonin and norepinephrine that keep some parts of the brain active while we are awake. Other neurons at the base of the brain begin signaling when we fall asleep. These neurons appear to "switch off" the signals that keep us awake. Research also suggests that a chemical called adenosine builds up in our blood while we are awake and causes drowsiness. This chemical gradually breaks down while we sleep (reference and reference). Here is a study based on switch on/switch off of fear i mice (reference).

There are also artificial ways in which to shut down receptors like our sweet receptors (reference). There have also been studies done on other organisms where one particular study showed the switch on/switch off of olfactory receptors in cockroach antenna (reference).