A quick hunt on Google Books turns up a few suggestions:
Roman Baths and Bathing(DeLaine & Johnson, Journal of Roman Archaeology 1999) looks like the book you need. The snippet claims that 'If Rome had a population of 0.9 million, its baths would need about 40,000 tonnes, making a total for Italy of 120,000' I would guess this figure is per annum...
The chapter on 'The Consumption and Cost of Fuel in Hypocaust Baths' looks like it provides more detailed estimates.
The common estimate of 114 tons/year for an average bath appears to derive from John Perlin's 1989 book
A Forest Journey. Another book (
Deforesting the Earth) relates this figure to the Welwyn baths, and says that the figure represents the equivalent of an acre of mature hardwood or 23 acres of coppiced forest.
The same book goes on to mention Severus Alexander setting aside woodlands to provide bath fuel, the rise of
lignari as specialist firewood traders, and salt contractors shipping firewood up the Tiber from North Africa.
This one also mentions a 'guild of wood suppliers' (presumably the
lignari again) with 60 ships used for importing bath wood. No reference appears to be given though!