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I can only speculate.

I think wood itself contains little phosphorus, compared to other parts of the plant. The inner rings of a tree are dead cells, a formerly living tissue, which condensed to cellulose and lignin. I assume the plant will not have phosphorus left there substantially. Phosphorus is needed to make ADP and DNA, by all living things, and is used in photosynthesis by plants, too. Quite useless in the middle of the dead wood zone. In woody plants, only the outer layers and leaves are living cells. If you leave those in the forest to rot, you retain some phosphorus.

You could also burn furniture after a century of use for energy and then bring the ash back into the forest to close the cycle, I guess. But that's requires non-toxic ashes - compatible paints and glues, no heavy metal agents and so on.

Either way, you are in competition with agriculture land use for food production. And bio fuels. And bio polymers. And bio... You get the idea.

Tho, if I had to guess, I would say wood probably is not the worst in terms of phosphorus leakage.