We present a targeted search for continuous gravitational waves (GWs) from 236 pulsars using data from the third observing run of LIGO and Virgo (O3) combined with data from the second observing run (O2). Searches were for emission from the l = m = 2 mass quadrupole mode with a frequency at only twice the pulsar rotation frequency (single harmonic) and the l = 2, m = 1, 2 modes with a frequency of both once and twice the rotation frequency (dual harmonic). No evidence of GWs was found, so we present 95\% credible upper limits on the strain amplitudes h0 for the single-harmonic search along with limits on the pulsars' mass quadrupole moments Q22 and ellipticities ε. Of the pulsars studied, 23 have strain amplitudes that are lower than the limits calculated from their electromagnetically measured spin-down rates. These pulsars include the millisecond pulsars J0437−4715 and J0711−6830, which have spin-down ratios of 0.87 and 0.57, respectively. For nine pulsars, their spin-down limits have been surpassed for the first time. For the Crab and Vela pulsars, our limits are factors of ∼100 and ∼20 more constraining than their spin-down limits, respectively. For the dual-harmonic searches, new limits are placed on the strain amplitudes C21 and C22. For 23 pulsars, we also present limits on the emission amplitude assuming dipole radiation as predicted by Brans-Dicke theory.