You get biased and inconsistent coefficient estimates, and biased standard errors. Bias in standard errors can be in both directions and the probability of types I and II errors could increase.
You can tackle non-linearity by introducing different functional forms of the predictor that had a non-linear relationship with Y. Common functional forms are quadratic, logarithmic, cubic, square roots, among others. You can also think about including splines and possibly interactions between two or more predictors. A last possibility is to use a different link function for the binary relationship, as functions such as probit and clog-log have slightly different shapes, albeit all of them following a synodal shape.