Rate-dependent tipping that concerns the effects of the rate of parameter change on the sudden transitions has been revealed in thermoacoustic systems. However, the conventional models cannot accurately portray the intermittent oscillations observed in experiments. This study explores the tipping behaviors in a thermoacoustic system with a secondary bifurcation, simultaneously accounting for the coupling effects of rate and colored noise. Particularly, the model contains higher-order nonlinearities, which lead to the emergence of both supercritical and subcritical bifurcations. It can qualitatively reproduce the intermittent dynamics of the system. We perform a transient analysis for the system via a stochastic averaging method and explore the influence mechanisms of colored noise and rate on the tipping phenomenon. The results show that the system exhibits a tipping phenomenon from the desired state to thermoacoustic instability through the state of intermittency. Interestingly, the rate causes the delay of tipping, while its increase enlarges the amplitude of intermittent oscillations. In addition, the system changes from an abrupt tipping to a smooth tipping for large noise intensities.