Using the Lyα mass assignment scheme (LyMAS), we make theoretical predictions for the 3-dimensional 3-point correlation function (3PCF) of the Lyα forest at redshift z = 2.3. We bootstrap results from the (100 h −1 Mpc) 3 Horizon hydrodynamic simulation to a (1 h −1 Gpc) 3 N -body simulation, considering both a uniform UV background (UVB) and a fluctuating UVB sourced by quasars with a comoving n q ≈ 10 −5 h 3 Mpc −3 placed either in massive halos or randomly. On scales of 10 − 30 h −1 Mpc, the flux 3PCF displays hierarchical scaling with the square of the 2PCF, but with an unusual value of Q ≡ ζ 123 /(ξ 12 ξ 13 + ξ 12 ξ 23 + ξ 13 ξ 23 ) ≈ −4.5 that reflects the low bias of the Lyα forest and the anti-correlation between mass density and transmitted flux. For halo-based quasars and an ionizing photon mean free path of λ = 300 h −1 Mpc comoving, UVB fluctuations moderately depress the 2PCF and 3PCF, with cancelling effects on Q. For λ = 100 h −1 Mpc or 50 h −1 Mpc, UVB fluctuations substantially boost the 2PCF and 3PCF on large scales, shifting the hierarchical ratio to Q ≈ −3. We scale our simulation results to derive rough estimate of the detectability of the 3PCF in current and future observational data sets for the redshift range z = 2.1−2.6. At r = 10 h −1 Mpc and 20 h −1 Mpc, we predict a signal-to-noise (SNR) of ∼ 9 and ∼ 7, respectively, for both BOSS and eBOSS, and ∼ 37 and ∼ 25 for DESI. At r = 40 h −1 Mpc the predicted SNR is lower by a factor of ∼ 3−5. Measuring the flux 3PCF would provide a novel test of the conventional paradigm of the Lyα forest and help separate the contributions of UVB fluctuations and density fluctuations to Lyα forest clustering, thereby solidifying its foundation as a tool of precision cosmology.