The combination of a flexible device and novel electromagnetic resonances offers new dimensions to manipulate electromagnetic waves and promises new device functionalities. In this study, we experimentally demonstrate a flexible metasurface that can support the bound state in the continuum (BIC) in the terahertz regime. The metasurface consists of toroidal dipole resonant units on top of the flexible polyimide substrate, which can support a terahertz Friedrich-Wintgen BIC resonance, and the resonance characteristics can be tuned by changing the parameters of the coupling unit among two resonant modes. The BIC resonances under different bending conditions are analyzed and compared, showing decent mechanical robustness. The sensing application is demonstrated by combining Fetal Bovine Serum with the flexible BIC metasurface. The measured minimum detectable concentration is 0.007 mg mL-1. Benefiting from the mechanical flexibility and BIC resonance characteristics, our approach can effectively manipulate terahertz waves and have potential applications in the realization of multifunctional and flexible photonic devices.