We study the Lagrangian intersection Floer theory of orbifold Riemann surfaces. First, we consider the Riemann sphere with four nontrivial orbifold singularities. There are two important Lagrangian submanifolds and we investigate the Lagrangian intersection Floer theory of them. The noncommutative mirror of orbifold Riemann sphere is given by a pair of noncommutative algebra and its central element obtained by the Floer theory. The other type of orbifold Riemann surfaces are coming from the Landau-Ginzburg orbifold which is a pair of polynomial and appropriate symmetry group which induces a group action on the level set. A quotient of the regular level set becomes the orbifold Riemann surface with boundary. For a given LG orbifold, we can define its dual Landau-Ginzburg orbifold algebraically. We give an interpretation of this duality in the language of mirror symmetry. In that sense, we construct the functor from A-model of LG orbifold to the B-model of its dual. We find a set of noncompact Lagrangian submanifolds and calculate the images under the functor which are their Floer complex with the Seidel Lagrangian in the quotient. They correspond to the set of all indecomposable matrix factorizations. This observation describes a geometric version of Auslander-Reiten theory.