We propose and demonstrate experimentally an electro-optic (EO) and thermo-optic (TO) tunable wavelength filter with band-rejection and band-pass dual-function. Our proposed filter is based on a long-period waveguide grating (LPWG) formed on a lithium niobate on insulator (LNOI) rib waveguide with a channel-shaped polymer cladding waveguide. The LPWG formed on the surface of the LNOI core enables efficient mode coupling between the two fundamental modes of the LNOI waveguide and the polymer cladding waveguide and hence dual-function filtering. For the z-/x-polarized input light, our fabricated best filter shows an 18.1-dB/8.2-dB band-rejection contrast and a 15.7-dB/17.4-dB band-pass contrast at 1565.26/1594.24 nm wavelength, with a relatively narrow 3-dB bandwidth of 2.5/2.3 nm, respectively. In addition, the fabricated filter also shows an EO tuning efficiency of ∼32 pm/V and a TO tuning efficiency of 0.465 nm/°C, respectively. Our proposed filter could find applications in wavelength division multiplexing, temperature sensing, and other fields of optical communication and optical information processing.