Metaphorically thinking, if one endured the pain of constant burning for decades, would the pain slowly lose its strength?
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).
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.
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.
Chronic pain may be very persistent and even worsen over time, but it will depend on the underlying pathology. If the pathology disappears over time, so may the pain.
There are roughly two kinds of pain:
Acute pain is pain that lasts for a short time and occurs following surgery or trauma or other condition. It acts as a warning to the body to seek help.
Chronic pain is pain that lasts beyond the time expected for healing healing following surgery or trauma or other condition. It can also exist without a clear reason at all.
A large body of basic research indicates that chronic pain is associated with neuroplastic changes in the nervous system at peripheral, spinal cord and brain levels. Thus chronic pain is shown to have a distinct pathology that often worsens over time, and constitutes a serious separate disease entity.
However, if the underlying pathology disappears, the pain may lessen. However, based on the cited paper, I do not think chronic pain gets less severe over time due to neural adaptation.
Excerpt from the linked article upon request:
These and other changes may contribute to the development of alterations in central [spinal] neuronal responsiveness, which share features of wind-up and long-term potentiation and are collectively known as “central sensitization.” [...] These alterations in cell responsiveness may be evidenced by a number of symptoms and signs, including spontaneous pain [...]. This [...] has been demonstrated in patients who have chronic pain associated with whiplash injury and fibromyalgia and suggests that the presence of pain [...] is associated with increased sensitivity of spinal neurons.
- Siddall & Cousins, Anesth Analg (2004); 99: 510-520.