If one endures the same form pain over a long period of time, would the pain begin to lose intensity?
Under the assumption that the temperature experienced is not high enough to simply burn the nerves responsible for the pain (which would lead to insensible regions), yes and it would due to what is called receptor desensitization. In the case of burning pain the receptor involved is called the transient-receptor potential channel V1 (TRPV1). It is the same receptor involve in the oral burning sensation of spiciness.

Now two scenarios are possible given your question, namely a constant stimulus over a long period and "oscillating" stimuli (i.e. a repetition of stimuli followed by non-stimulated periods).

Constant stimulus

I provided a similar answer to this question: What features cause mechano sensory adaptation?. This is the scenario where you keep the painful stimulus constant over a period of time. Here desensitization, i.e. reduction in pain feeling, is due to a direct receptor desensitization. This happens because the receptor is internalized (also called receptor-mediated endocytosis) into the cell after being activated and it takes some times for the receptor to reach the cell membrane again meaning if the stimulus remains constant, pretty much no receptors will be left on the cell membrane.  After decades long term desensitization will occur and is slightly different in it's mechanism compared to standard desensitization as described hereafter.

"Oscillating" stimuli

In the case you are talking about a chronic pain interrupted by periods of non stimulation then long term desensitization will occur. In this scenario, the concentration of receptors will be reduced semi-permanently (i.e. reduced expression of the receptor) in the receptor cells reducing the pain signal. This is what happens with spiciness sensation. People eating regularly very spicy will physiologically be less sensitive to spices as they have less receptors sensing the spices provoking a reduced burning sensation.



Just as a footnote, very long chronic pain will also change the brain wiring of the pain signal but as I don't know enough about neurobiology I will leave that appart.