Searching topological states in artificial systems has recently become a rapidly growing field of research. Meanwhile, significant experimental progress on observing topological phenomena has been made in superconducting circuits. However, topological insulator states have not yet been reported in this system. Here, for the first time, we experimentally realize a tunable dimerized spin chain model and observe the topological magnon insulator states in a superconducting qubit chain. Via parametric modulations of the qubit frequencies, we show that the qubit chain can be flexibly tuned into topologically trivial or nontrivial magnon insulator states. Based on monitoring the quantum dynamics of a single-qubit excitation in the chain, we not only measure the topological winding numbers, but also observe the topological magnon edge and defect states. Our experiment exhibits the great potential of tunable superconducting qubit chain as a versatile platform for exploring noninteracting and interacting symmetry-protected topological states.