Dynamin proteins assemble into characteristic helical structures around necks of clathrin-coated membrane buds. Hydrolysis of dynamin-bound GTP results in both fission of the membrane neck and partial disruption of the dynamin oligomer. Imaging by atomic force microscopy reveals that, on GTP hydrolysis, dynamin oligomers undergo a dynamic remodeling and lose their distinctive helical shape. While breakup of the dynamin helix is a critical stage in clathrin-mediated endocytosis, the mechanism for this remodeling of the oligomer has not been resolved. In this paper, we formulate an analytical, elasticity-based model for the reshaping and disassembly of the dynamin scaffold. We predict that the shape of the oligomer is modulated by the orientation of dynamin's pleckstrin homology (PH) domain relative to the underlying membrane. Our results indicate that tilt of the PH domain drives deformation and fragmentation of the oligomer, in agreement with experimental observations. This model motivated the introduction of the tilted helix: a curve that maintains a fixed angle between its normal and the normal of the embedding surface. Our findings highlight the importance of tilt as a key regulator of size and morphology of membrane-bound oligomers.