YES! You sound right on target there with burning it down! I would then treat the area with lime and diatomaceous earth (Also just called DE).
The following I gleaned off the web so I can't take credit for it but it's good information;
"Bird mites" or "Tropical fowl mites" are the common names used to describe the mite Ornithonyssus bursa from the family of mites Macronyssidae; these mites are often incorrectly called ‘bird lice’, particularly within the pest control industry, and are also commonly the cause of ‘paper mite’ problems (strictly speaking there are no such things as ‘paper mites’). This species of bird mite is widely distributed throughout warmer regions of the world. The mites are haematophagous (feed on blood) natural parasites of common birds including pigeons, starlings, sparrows, Indian mynahs, poultry, and some wild birds. Ornithonyssus bursa mites are small with eight legs, barely visible to the eye, oval in shape with a sparse covering of short hairs, and are extremely mobile. They are semitransparent in color which makes them difficult to detect on skin until blood is ingested and then digested - when they may appear reddish to blackish.
The mites remaining in the abandoned nest, and left without a suitable host when the birds leave or die, will roam and disperse throughout the dwelling over a 1-2 week period searching for new hosts. Most mites will die within 10 days without a blood meal from a bird host. They will bite but cannot survive on humans and do not infect/infest human skin.