Are there subexponential algorithms for PLANAR SAT known? Some NP-hard problems which are exponential on general
graphs are subexponential on planar graphs because the
treewidth is at most $4.9 \sqrt{|V(G)|}$ and they are exponential
in the treewidth.

Basically I am interested if there are subexponential
algorithms for PLANAR SAT which is NP-complete.

Let $\phi$ be a CNF formula on variables $x_i$ and the
$i$-th clause is $c_i$.

The incidence graph p. 5
$G$ of $\phi$ is on vertices $V(G)=\{x_i\} \cup \{c_i\}$
and edges $(x_i,c_i)$ iff $x_i \in c_i$ or $\lnot x_i \in c_i$.

$\phi$ is in PLANAR SAT if the incidence graph is planar.


  Are there subexponential algorithms for PLANAR SAT in terms of $\phi$?


I don't exclude the possibility reduction SAT to PLANAR SAT
to make this possible, though SAT still to be exponential
and $\phi$ is subexponential because of the increase in the size.
Well, you can apply the planar separator theorem together with dynamic programming and get running time $2^{O(\sqrt{n})}$, where $n$ is the number of vertices in the graph. The idea being that you try all possible assignments for the variable vertices on the separator, and all variables mentioned in clauses in the separator (assuming each clause has a constnat number of variables). 

If a clause node is large, then you have to be a bit more clever - you have to guess whether to assign it to the left side or right side subproblem. The details for such things tends to be messy and not immediate, so I am not going to give more details. I think the original papers by Lipton and Tarjan solved similar problems using similar ideas, if my memory serves me right.