  Precision tests of general relativity can be conducted by observing binary
pulsars. Theories with massive fields exist to explain a variety of phenomena
from dark energy to the strong CP problem. Existing pulsar binaries, such as
the white dwarf-pulsar binary J1738+0333, have been used to place stringent
bounds on the scalar dipole emission, and radio telescopes may detect a pulsar
orbiting a black hole in the future. In this paper, we study the ability of
pulsar binaries to probe theories involving massive scalar and vector fields
through the measurement of the orbital decay rate. With a generic framework, we
describe corrections to orbital decay rate due to (a) modification of GR
quadrupolar radiation and (b) dipolar radiation of a massive field. We then
consider three concrete examples: (i) massive Brans-Dicke theory, (ii) general
relativity with axions, and (iii) general relativity with bound dark matter and
a dark force. Finally, we apply direct observations of J1738 and simulations of
a black hole-pulsar binary to bound theory parameters such as field's mass and
coupling constant. We find new constraints on bound dark matter interactions
with PSR J1738, and a black hole-pulsar discovery would likely improve these
further. Such bounds are complementary to future gravitational-wave bounds.
Regarding other theories, we find similar constraints to previous pulsar
measurements for massive Brans-Dicke theory and axions. These results show that
new pulsar binaries will continue to allow for more stringent tests of gravity.
