Several years ago I wrote a number of posts about Logit and Probit models, and the Linear Probability Model LPM). One of those posts (also, see here) dealt with the problems that arise if you mis-classify the dependent variable in such models. That is, in the binary case, if some of your "zeroes" should be "ones", and/or vice versa.
In a conventional linear regression model, measurement errors in the dependent variable are not a biog deal. However, the situation is quite different with Logit, Probit, and the LPM.
This issue is taken up in detail in an excellent, recent, paper by Meyer and Mittag (2017), and I commend their paper to you.
To give you an indication of what those authors have to say, this is from their Introduction:
".....the literature has established that misclassification is pervasive and affects estimates, but not how it affects them or what can still be done with contaminated data. This paper characterizes the consequences of misclassification of the dependent variable in binary choice models and assesses whether substantive conclusions can still be drawn from the observed data and if so, which methods to do so work well. We first present a closed form solution for the bias in the linear probability model that allows for simple corrections. For non-linear binary choice models such as the Probit model, we decompose the asymptotic bias into four components. We derive closed form expressions for three bias components and an equation that determines the fourth component. The formulas imply that if misclassification is conditionally random, only the probabilities of misclassification are required to obtain the exact bias in the linear probability model and an approximation in the Probit model. If misclassification is related to the covariates, additional information on this relation is required to assess the (asymptotic) bias, but the results still imply a tendency for the bias to be in the opposite direction of the sign of the coefficient."
This paper includes a wealth of information, including some practical guidelines for practitioners.
Meyer, B. D. and N. Mittag, 2017. Misclassification in binary choice models. Journal of Econometrics, 200, 295-311.