The behavior of linear polymer chains in dilute solution flows has an established history. Polymers often possess more complex architectures, however, such as branched, dendritic, or ring structures. A major challenge lies in understanding how these nonlinear chain topologies affect the dynamic properties in nonequilibrium conditions, in both dilute and entangled solutions. In this work, we interrogate the single-chain dynamics of ring polymers using a combination of simulation, theory, and experiment. Inspired by recent experimental results by Li et al., we demonstrate that the presence of architectural constraints has surprising and pronounced effects on the dynamic properties of polymers as they are driven out of equilibrium. Ring constraints lead to two behaviors that contrast from linear chains. First, the coil-stretch transition occurs at larger values of the dimensionless flow strength (Weissenberg number) compared to linear chains, which is driven by coupling between intramolecular hydrodynamic interactions (HI) and chain architecture. Second, a large loop conformation is observed for ring polymers in extensional flow at intermediate to large Weissenberg numbers, and we show that this open loop conformation is driven by intramolecular HI. Our results reveal the emergence of new paradigms in chain architecture hydrodynamic coupling that may be relevant for solution-based processing of polymeric materials and could provide new opportunities for precise flow-based polymer conformation control to guide material properties.