We performed near-daily observations on the binary pulsars PSR J0218+4232, PSR J1518+4904 and PSR J2023+2853 with the Canadian Hydrogen Intensity Mapping Experiment (CHIME). For the first time, we detected the Shapiro time delay in all three pulsar-binary systems, using only 2--4 years of CHIME/Pulsar timing data. We measured the pulsar masses to be \(1.49^{+0.23}_{-0.20}\) M\(_\odot\), \(1.470^{+0.030}_{-0.034}\) M\(_\odot\) and \(1.50^{+0.49}_{-0.38}\) M\(_\odot\) respectively. The companion mass to PSR J0218+4232 was found to be \(0.179^{+0.018}_{-0.016}\) M\(_\odot\). We constrained the mass of the neutron-star companion of PSR J1518+4904 to be \(1.248^{+0.035}_{-0.029}\) M\(_\odot\), using the observed apsidal motion as a constraint on mass estimation. The binary companion to PSR J2023+2853 was found to have a mass of \(0.93^{+0.17}_{-0.14}\) M\(_\odot\); in the context of the near-circular orbit, this mass estimate suggests that the companion to PSR J2023+2853 is likely a high-mass white dwarf. By comparing the timing model obtained for PSR J0218+4232 with previous observations, we found a significant change in the observed orbital period of the system of \(\dot{P_{\rm b}} = 0.14(2) \times 10^{-12}\); we determined that this variation arises from ``Shklovskii acceleration" due to relative motion of the binary system, and used this measurement to estimate a distance of \(d=(6.7 \pm 1.0)\) kpc to PSR J0218+4232. This work demonstrates the capability of high-cadence observations, enabled by the CHIME/Pulsar system, to detect and refine general-relativistic effects of binary pulsars over short observing timescales.