  We present Herschel observations of far-infrared (FIR) fine-structure (FS)
lines [CII]158$\mu$m, [OI]63$\mu$m, [OIII]52$\mu$m, and [SiII]35$\mu$m in the
z=2.56 Cloverleaf quasar, and combine them with published data in an analysis
of the dense interstellar medium (ISM) in this system. Observed [CII]158$\mu$m,
[OI]63$\mu$m, and FIR continuum flux ratios are reproduced with
photodissociation region (PDR) models characterized by moderate far-ultraviolet
(FUV) radiation fields $G_0=$ 0.3-1$\times10^3$ and atomic gas densities
$n_{\rm H}=$ 3-5$\times10^3$ cm$^{-3}$, depending on contributions to
[CII]158$\mu$m from ionized gas. We assess the contribution to [CII]158$\mu$m
flux from an active galactic nucleus (AGN) narrow line region (NLR) using
ground-based measurements of the [NII]122$\mu$m transition, finding that the
NLR can contribute at most 20-30% of the observed [CII]158$\mu$m flux. The PDR
density and far-UV radiation fields inferred from the atomic lines are not
consistent with the CO emission, indicating that the molecular gas excitation
is not solely provided via UV-heating from local star-formation, but requires
an additional heating source. X-ray heating from the AGN is explored, and we
find that X-ray dominated region (XDR) models, in combination with PDR models,
can match the CO cooling without overproducing observed FS line emission. While
this XDR/PDR solution is favored given the evidence for both X-rays and
star-formation in the Cloverleaf, we also investigate alternatives for the warm
molecular gas, finding that either mechanical heating via low-velocity shocks
or an enhanced cosmic-ray ionization rate may also contribute. Finally, we
include upper limits on two other measurements attempted in the Herschel
program: [CII]158$\mu$m in FSC~10214 and [OI]63$\mu$m in APM~08279+5255.
