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Essentially, this question could also be asked: how many species do all of the paleontological specimens in collections (or collections records) represent?
I recognize that species is a temporal concept when thinking about differentiation across time; but, natural history collection catalog specimens to species level, so there must be a reasonable basis to make this distinction. Also, note that I'm not exactly interested in inferred species (i.e. if we have records of species A and species B, we might infer a last common ancestor, species C).
Very interesting question.
I remember reading about it in one of Stephen Jay Gould's great books (The Panda's Thumb? The Flamingo's Smile? Wonderful Life: The Burgess Shale and the Nature of History? Dinosaur in a Haystack? Probably in one of these).
He wrote exactly about this question, and the effort made by teams from many different countries to get to an "approximate" result. What really impressed me at the time (fifteen years ago) is that when Chinese scientists were invited to add material for a second analysis, the number of known fossil species doubled!
Please, if anyone find the exact source of this, I'll be glad to read it!
But maybe you can get an idea by looking at the number of fossil species for certain groups, and comparing with the number of estimated/known extant species. Keep in mind the rates of fossilization for each group, and also the fact that Nematoda and Acari might be the most overlooked groups in terms of species number (both seem to parasitize animals through most of the Animal Kingdom).
Short answer no one knows exactly, a single database of fossils does not exist. there are lots of specialized ones, for say pollen or dinosaurs, but not one complte one. each museum maintains their own. There are estimates based on type specimen publications but they tend to overemphasize some groups like vertebrates and under represent less… charismatic groups. Accroding to this study http://www.pnas.org/content/99/6/3706.full There are at least 280,000 named invertibrate species, (infered species are not given species names) hopefully that gets you started. This is a recognized problem and there is a lot of debate about how to fix it.