Simple answer - yes. Permititis afflicts a number of dales caves. As Simon notes, not everyone worries about this too much....
The Council of Northern Caving Clubs covers most of them - requiring permits for lots of holes on PenyGhent/Fountains Fell, for Leck Fell and for Easegill and others. Easegill is pretty relaxed but the landowner at Leck Fell got a bit snotty when the world and his dog started pirating NottsII, and Fountains Fell is pretty restricted
Gaping Gill and The Allotment also need permits, these are actually pretty good as GG permits will be issued the day before over the phone by the landowner - it just reduces the cave knitting with 20 groups down there.
The CNCC handbook gives the full details of how to get permits for each cave - until/unless you know the local politics it's probably best to get permits for everything that "needs" one - not worth upsetting landowners and getting it shut for everyone.