We study the noisy voter model using a non-linear dependence of the rates that mimics collective interaction between individuals. The resulting model is solved exactly under the all-to-all coupling configuration and approximately in some random networks environments. In the all-to-all setup we find that the non-linear interactions induce bona fide phase transitions that, contrarily to the linear version of the model, survive in the thermodynamic limit. For sufficiently large values of the non-linearity parameter, first and second transition lines cross in a tricritical point and tristability is reported in a region of the parameter space. All the results are generalized to complex networks with the help of the pair approximation. The main effect of the complex network is to shift the transition lines and modify the finite-size dependence, a modification that can be captured with the introduction of an effective system size that decreases with the degree heterogeneity of the network. These theoretical predictions are well confirmed by numerical simulations of the stochastic process.