Are there sites where shrubs and perennial grasses and forbs can persist beneath closed pinyon and/or juniper canopies?

Yes and no.  Shrubs rarely persist beneath a closed tree canopy, especially sagebrush.  However, deep rooted perennial grasses have been observed to persist on sites with deep soils and where restrictive layers (e.g. clay accumulation layer called an argillic), reducing water capture and storage are absent or weakly developed.  Photos of two different pinyon-juniper sites, the first (top) with moderately deep to deep soils with no restrictive layer and the second (bottom) with soils having a shallow restrictive layer.

Both sites have long since lost the shrub layer (evidence of old shrub skeletons) but the perennial herbaceous layer persists where soils are moderately deep to deep with no or limited restrictive layer. Where the restrictive layer is well developed the herbaceous layer is typically severely depleted. (Photos by Rick Miller).