Quantum critical systems constitute appealing platforms for the exploration of novel measurement-induced phenomena due to their innate sensitivity to perturbations. We study the impact of measurement on Ising chains using an explicit protocol, whereby uncorrelated ancillae are entangled with the critical chain and then projectively measured. These measurements can modify the Ising order-parameter scaling dimension in a significant way, and we show that varying the measurement basis can induce a flow towards different conformal invariant boundary conditions. We validate this analysis for two different microscopic realizations of the same critical theory. We further investigate the impact of measurements on the tricritical Ising model, finding how they can trigger a non-trivial flow to different boundary conditions.