We develop a resilient distributed strategy over multitask networks, where individual tasks are linearly related within each neighborhood, and information exchanges between neighboring agents are noisy. In the proposed strategy, each agent follows an adapt-then-project procedure to iteratively update its local estimate. In particular, weighted projection operators are utilized in the projection step in order to attenuate the negative effect of noisy exchanges on the cooperative inference performance. We motivate a strategy for computing the weights in a distributed and adaptive manner. Simulation results demonstrate that the proposed scheme shows good resilience against noise in the information exchange between agents.