I have already talked about my objection to burning on allotments in my blog "My Rant About Garden Bonfires".
However, I have thought of another reason for burying woody material in the subsoil rather than burning. There is nothing new and when used to make raised beds is called Hugelkultur. All I would add about this is that burying woody material will aid in drainage.
I accept that placing woody material in soil depletes it of nitrogen. Bacteria and particularly fungi take in nitrogen in order to break down cellulose and lignin. If there is free nitrogen, as in the anion NO3- indeed it will be rejected by the negative charge on most organic matter and soil particles and be leached out of the soil. However the foraging hyphae of fungi decomposing woody material may well mop up any NO3 - that is available in the soil when decomposing brush litter.
It seems that size of the brushwood fragments, burial of remains and nitrogen addition positively influences fungal and bacterial biomass and activity. (ref: van der Wal, De Boer, Smant, and van Veen (2007). I would suggest that the increased activity of these organisms is brought about by absorption of leached soil NO3- after the nitrogen sources within the decaying vegetation have been depleted. The fact that there is not lots of undecomposed woody material scattered throughout the soil profile would suggest that decomposition does occur, albeit slower than in the top soil, and that nitrogen for this process must be coming from somewhere. I conjecture that leached NO3- from the top soil is a good contender for this nitrogen.
If we can trap this nitrogen within the bodies of bacteria and fungi then there is more of a chance of recycling it later when this material is returned towards the surface of the soil. Conversely, it may become available to plant roots or associated mychorrhizal fungi hyphae. Regardless, nitrogen will be recycled rather than being leached away and it passing into water courses.