Robots planning long-horizon behavior in complex environments must be able to quickly reason about the impact of the environment's geometry on what plans are feasible, i.e., whether there exist action parameter values that satisfy all constraints on a candidate plan. In tasks involving articulated and movable obstacles, typical Task and Motion Planning (TAMP) algorithms spend most of their runtime attempting to solve unsolvable constraint satisfaction problems imposed by infeasible plan skeletons. We developed a novel Transformer-based architecture, PIGINet, that predicts plan feasibility based on the initial state, goal, and candidate plans, fusing image and text embeddings with state features. The model sorts the plan skeletons produced by a TAMP planner according to the predicted satisfiability likelihoods. We evaluate the runtime of our learning-enabled TAMP algorithm on several distributions of kitchen rearrangement problems, comparing its performance to that of non-learning baselines and algorithm ablations. Our experiments show that PIGINet substantially improves planning efficiency, cutting down runtime by 80% on average on pick-and-place problems with articulated obstacles. It also achieves zero-shot generalization to problems with unseen object categories thanks to its visual encoding of objects.