  We study the tropical Dolbeault cohomology of non-archimedean curves as
defined by Chambert-Loir and Ducros. We give a precise condition for when this
cohomology satisfies Poincar\'e duality. The condition is always satisfied when
the residue field of the non-archimedean base field is the algebraic closure of
a finite field. We also show that for curves over fields where the residue
field is the field of complex numbers, the tropical (1,1)-Dolbeault cohomology
can be infinite dimensional. Our main new ingredient is an exponential type
sequence that relates tropical Dolbeault cohomology to the cohomology of the
sheaf of harmonic functions. As an application of our Poincar\'e duality
result, we calculate the dimensions of the tropical Dolbeault cohomology,
called tropical Hodge numbers, for (open subsets of) curves.
