A label-free optical microring resonator biosensor based on lithium niobate-on-insulator (LNOI) technology is designed and simulated for biosensing applications. Although silicon-on-insulator technology is quite mature over LNOI for fabricating more compact microring resonators, the latter is attractive for its excellent electro-optic, ferroelectric, piezoelectric, photoelastic, and nonlinear optic properties, which can offer a wide range of tuning facilities for sensing. To satisfy the requirement of high sensitivity in biosensing, the dual-microring resonator model is applied to design the proposed sensor. The transmission spectrum obtained from two-dimensional simulations based on finite-difference time-domain method demonstrates that the designed LNOI microring sensor consisting of a outer ring and a inner ring offers a sensitivity of index unit (RIU) and a minimum detection limit of RIU. Finally, the sensor’s performance is simulated for glucose sensing, a biosensing application.