Decoherence inevitably happens when a quantum state is exposed to its environment, which can affect quantum critical points (QCPs) in a nontrivial way. As was pointed out in the recent literature on (1+1)d conformal field theory (CFT) [Garratt et al. Measurements conspire nonlocally to restructure critical quantum states, arXiv:2207.09476 (2022)], the effect of weak measurement can be mathematically mapped to the problem of boundary CFT. In this work, we focus on the (2+1)d QCPs, whose boundary and defect effects have attracted enormous theoretical and numerical interests very recently. We focus on decoherence caused by weak measurements with and without postselecting the measurement outcomes. Our main results are as follows: (1) for an O(N) Wilson-Fisher QCP under weak measurement with postselection, an observer would, in general, observe two different types of boundary and defect criticality with very different behaviors from the well-known Wilson-Fisher fixed points; in particular, it is possible to observe the recently proposed exotic "extraordinary-log"correlation. (2) An extra quantum phase transition can be driven by decoherence, if we consider quantities nonlinear with the decohered density matrix, such as the Renyi entropy. We demonstrate the connection between this transition to the information-theoretic transition driven by an error in the toric code model. (3) When there is no postselection, though correlation functions between local operators remain the same as the undecohered pure state, nonlocal operators such as the "disorder operator"would have qualitatively distinct behaviors; and we also show that the decoherence can lead to confinement. © 2023 authors. Published by the American Physical Society. Published by the American Physical Society under the terms of the "https://creativecommons.org/licenses/by/4.0/"Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.