What would happen if brain neurons did not reduce their sensitivity to neurotransmitters after prolonged exposure?
Short answer
As far as I am aware, neurotoxic effects of chronic excitatory stimulation are not prevented by receptor trafficking. 

Background
I think the question assumes that 1) neural damage related to continuous activation (excitotoxicity) is preventable by 2) the dynamic regulation of receptor numbers on the neuronal membrane. As far as I can see, these two processes are unrelated. 


Excitotoxicity is 


  ...the ability of glutamate or related excitatory amino acids to mediate the death of central neurons [...] for example, after intense exposure. [...] Excitotoxicity has substantial cellular specificity and, in most cases, is mediated by glutamate receptors. On average, NMDA receptors activation may be able to trigger lethal injury more rapidly than AMPA or kainate receptor activation, perhaps reflecting a greater ability to induce calcium influx and subsequent cellular calcium overload (Choi, 1992).      



Hence, it is the influx of toxic amounts of Ca2+ that induce cell death. Not much Ca2+ is needed to accomplish this. For example, a status epilepticus (a sustained epileptic seizure) can lead to hippocampal damage, due to persistent excitation and glutamate release (Meldrum, 1991). Epileptic seizures lasting longer than 5 minutes are regarded as dangerous in this sense and after 30 minutes they are potentially fatal (often not directly related to excitotoxicity though) (Cherian &amp; Thomas, 2009). In this short time span regulation of cell-surface receptor densities are unlikely to play a substantial role. For example, serotonin-specific reuptake inhibitors need 2 weeks to lower the amount of 5HT1A receptors (Tylee &amp; Wolters, 2009).         


Dynamic regulation of receptors. The regulation of the number of receptors is linked via feedback loops. A great example is the way glutamatergic AMPA receptors are regulated through endocytosis (Fig. 1).



Fig. 1. AMPA receptor trafficking. Source: Anggono Lab. 

AMPA receptors are the principal receptors that mediate most of the fast excitatory synaptic transmission in the mammalian brain (Purves et al., 2001). The dynamic trafficking and proper synaptic targeting of AMPA receptors are crucial in determining the strength and plasticity of excitatory synaptic transmission. This dynamic regulation may also impact learning and memory processes. The process of receptor trafficking is, as far as I know, typically not associated with emergency rescue operations to prevent excitotoxicity. Nonetheless, these mechanisms, are essential to sustain an adaptable brain capable of learning, forgetting and restructuring itself. 

References
- Cherian &amp; Thomas, Ann Indian Acad Neurol (2009); 12(3): 140–53
- Choi, J Neurobiol (1992); 23(9):1261-76
- Meldrum, Epilepsy Res (1991); 10(1): 55-61
- Purves et al., Neuroscience. 2nd ed. Sunderland (MA): Sinauer Associates; 2001
- Tyllee &amp; Wolters, BMJ (2007); 334(7600): 911–2