Concerning fossil tunicates and whether they are known from non-marine strata you could ask Mike Reich and colleagues who have been working on ascidian spicules:
there are no freshwater tunicates at all … only a few species can tolerate salinities below 22‰ … this is connected with the development and physiology, tunicate larvae as well as the zooids in adult specimens will affected dramatically by freshwater resp. brackish water … presumably, this is also connected/related to the unique biomineralisation within tunicates, here you can find for instance weird/unusal biominerals like vaterite or oxalates …
Concerning the salinity of tunicates you can check:
Lambert, Gretchen (2005): Ecology and natural history of the protochordates. Canadian Journal of Zoology 83(1): 34-50.
Sims, Linda L. (1984): Osmoregulatory capabilities of three macrosympatric stolidobranch ascidians, Styela clava Herdman, S. plicata (Lesueur) and S. montereyensis (Dall). Journal of Experimental Marine Biology and Ecology 82(2-3): 117-129.
Even in the (rare) fossil record of tunicates, there are only findings from marine/stenohaline environments known...
Thank you, Mike Reich, for your reply. Why are tunicate osmoregulatory capabilities so poor in comparison to, say, vertebrates? With Echinodermata, this seems to make sense (they use seawater for blood), but this constraint would not seem to apply to ascidians.
no freshwater tunicates, but they can be found in lower salinity water. see article below. http://charlotte.ifas.ufl.edu/seagrant/publications/EAS_060413_006_Tunicates.pdf
this paper gives some clue as to why most of them do not osmoregulate well.