The reasons why linear regressions cannot be used in case of binary classification are as follows:
Distribution of error terms: The distribution of data in the case of linear and logistic regression is different. Linear regression assumes that error terms are normally distributed. In the case of binary classification, this assumption does not hold true.
Model output: In linear regression, the output is continuous. In the case of binary classification, an output of a continuous value does not make sense. For binary classification problems, linear regression may predict values that can go beyond 0 and 1. If we want the output in the form of probabilities, which can be mapped to two different classes, then its range should be restricted to 0 and 1. As the logistic regression model can output probabilities with logistic/sigmoid function, it is preferred over linear regression.
Variance of Residual errors: Linear regression assumes that the variance of random errors is constant. This assumption is also violated in the case of logistic regression.