Quantum measurements inevitably disturb the state being measured, and this disturbance, together with measurement incompatibility, plays a crucial role in sequential sharing of nonlocality. While the information-disturbance trade-off for a single measurement is well understood, its interplay with measurement incompatibility remains unexplored. Here, we investigate this interplay in the sequential sharing of three forms of tripartite nonlocality: standard, genuine, and nosignalling-genuine. By deriving incompatibility trade-off relations among the measurements performed by the three observers, we show that the role of measurement incompatibility differs qualitatively across three forms of nonlocality. Standard tripartite nonlocality can be demonstrated even when one party performs compatible measurements, rendering unbounded sequential sharing essentially trivial. In contrast, genuine tripartite nonlocality cannot be realised when any two parties perform nearly compatible measurements, precluding unbounded sharing. For nosignalling-genuine nonlocality, however, the incompatibility trade-off allows the critical measurement unsharpness to be arbitrarily small, along with another parties measurement to be almost compatible, so that the minimum measurement-induced disturbance required to demonstrate nonlocality can also be made arbitrarily small, preserving sufficient quantum correlations for unbounded sequential sharing. We further show that, in this regime, the feasibility of unbounded sequential sharing depends sensitively on the Kraus realisation implementing the measurement. Our results demonstrate that measurement incompatibility alone does not fully determine the limits of sequential multipartite nonlocality; instead, the minimum measurement-induced disturbance required to achieve nonlocality plays an equally decisive operational role.