We present observations of the classical T Tauri star DO Tau collected with the near-infrared SPIRou spectropolarimeter and precision velocimeter at the Canada-France-Hawaii Telescope from early 2020 to late 2025. Circularly polarized Zeeman signatures were clearly detected at most epochs in the atomic spectral lines of DO Tau, yielding longitudinal magnetic fields of up to 280 G modulated with a period of 5.128+-0.002 d which we identified as the rotation period of DO Tau. Applying Zeeman-Doppler imaging to the SPIRou data recorded in 2021, 2024 and 2025, we found that DO Tau hosts an unusual large-scale magnetic field that is weaker, less poloidal, more inclined to the rotation axis, and varies more rapidly with time than those of previously studied T Tauri stars, possibly as a result of intense accretion between the inner disk and the stellar surface. The dipole component of this large-scale field of about 0.2-0.3 kG even flipped polarity toward the end of our observing campaign, making DO Tau the first T Tauri star for which a magnetic polarity reversal is reported. The magnetospheric gap surrounding the central star was quite compact, extending to ~1.6 Rstar (0.014 au) as a result of the strong accretion rate (log Mdot = -7.7 Msun/yr), with the inner accretion disk being warped by the tilted stellar magnetic field. Radial velocity variations suggest the presence of a close-in planet of a few Mjup or a density structure in the inner accretion disk at an orbital period of 21 d (corresponding to 0.12 au), which might be linked to the wiggle in the jet axis of DO Tau.