F-BAR proteins link cellular membranes to the actin cytoskeleton in many biological processes. Here we investigated the function of the Schizosaccharomyces pombe Imp2 F-BAR domain in cytokinesis and find that it is critical for Imp2's role in contractile ring constriction and disassembly. To understand mechanistically how the F-BAR domain functions, we determined its structure, elucidated how it interacts with membranes, and identified an interaction between dimers that allows helical oligomerization and membrane tubulation. Using mutations that block either membrane binding or tubulation, we find that membrane binding is required for Imp2's cytokinetic function but that oligomerization and tubulation, activities often deemed central to F-BAR protein function, are dispensable. Accordingly, F-BARs that do not have the capacity to tubulate membranes functionally substitute for the Imp2 F-BAR, establishing that its major role is as a cell-cycle-regulated bridge between the membrane and Imp2 protein partners, rather than as a driver of membrane curvature.