In this article, we consider a class of unconstrained multiobjective interval-valued optimization problems in which each objective function is a sum of a generalized Hukuhara(gH)-differentiable interval valued function and a proper, convex and gH-lower semicontinuous interval valued function, which is possibly non-gH-differentiable. Within this framework, we introduce the notion of a proximal gradient direction and characterize the Pareto critical points of the considered problem through a strongly convex auxiliary subproblem. Based on these characterizations, we develop a proximal gradient algorithm with an Armijo-type line search that ensures sufficient descent with respect to the interval partial order. We then establish the existence of an uniform lower bound on the admissible step length, the global convergence of the generated sequence to a Pareto critical point followed by a global rate of convergence in the general nonconvex setting. The theoretical findings of the proposed method are validated through some test problems and further illustrate its applicability on portfolio selection problems involving interval-valued data.