Particle-like structures of solitons, as a result of the balance between dispersion and nonlinearity, enable remarkable elastic and inelastic soliton collisions in many fields. Despite the experimental observation of temporal vector-soliton collisions in birefringent fibers, collision dynamics of vector solitons in fiber lasers have not been revealed before, to the best of our knowledge. Here, the real-time spectral evolutions of vector solitons during collisions in a dual-comb fiber laser, which generates vector solitons with slightly different repetition rates, are captured by a time-stretch dispersive Fourier transform technique. We record the whole process of vector-soliton collisions, including the formation of weak pulses induced by cross-polarization coupling, opposite central wavelength shifts of both vector solitons, distinct intensity redistribution and dissipative energy, and gradual recovery to initial states. Furthermore, extreme collisions with strong four-wave mixing sidebands are observed by virtue of coherent coupling between the orthogonal polarization components of vector solitons. Numerical simulations match well with the experimental observations. The experimental and numerical evidences of vector-soliton collision dynamics could give insight into the understanding of nonlinear dynamics in fiber lasers and other physical systems, as well as the improvement of laser performance for application in dual-comb spectroscopy.