Outliers may have the same essential impact on a logistic regression as they have in linear regression: The deletion-diagnostic model, fit by deleting the outlying observation, may have DF-betas greater than the full-model coefficient; this means that the sigmoid-slope of association may be of opposite direction. Separately, the inference may not agree in the two models, suggesting one commits a type II error, or the other commits a type I error.
This point underscores the problem of suggesting that, when outliers are encountered, they should summarily be deleted.
The implication for logistic regression data analysis is the same as well: if there is a single observation (or a small cluster of observations) which entirely drives the estimates and inference, they should be identified and discussed in the data analysis. DF-beta residual diagnostics is an effective numerical and graphical tool for either type of model which is easy to interpret by statisticians and non-statisticians alike.
There are some differences to discuss. In linear regression, it is very easy to visualize outliers using a scatter plot. The scaled vertical displacement from the line of best fit as well as the scaled horizontal distance from the centroid of predictor-scale X together determine the influence and leverage (outlier-ness) of an observation. For a logistic model, the mean-variance relationship means that the scaling factor for vertical displacement is a continuous function of the fitted sigmoid slope. Farther out in the tails, the mean is closer to either 0 or 1, leading to smaller variance so that seemingly small perturbations can have more substantial impacts on estimates and inference. However, whereas a Y value in linear regression may be arbitrarily large, the maximum fitted distance between a fitted and observed logistic value is bounded. Does that mean that a logistic regression is robust to outliers? Absolutely not.