Is #HORNSAT polynomial?
Even if decision is easy, the counting problem #HORNSAT is #P-complete [1], the counting analog of NP-completeness problem. Thus it is very unlikely it has a polynomial time algorithm (for example, if $P \neq NP$ then #HORNSAT has no ptime algorithm). 

See [1] for a dichotomy characterizing the clause restrictions leading to tractable instances of #SAT. More precisely, it is shown that the only restriction of clauses that gives tractable instances for #SAT are the clauses of the form $x_1 \oplus x_2 \oplus ...\oplus x_n = 0$ where $\oplus$ is the sum modulo $2$. The other restrictions (#2-SAT, #HORNSAT, ...) are all #P-complete.

[1] Nadia Creignou, Miki Hermann: Complexity of Generalized Satisfiability Counting Problems. Inf. Comput. 125(1): 1-12 (1996), http://www.sciencedirect.com/science/article/pii/S0890540196900164?via%3Dihub