Are there subexponential algorithms for PLANAR SAT known?
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.