Is multicollinearity an issue when doing stepwise logistic regression using AIC and BIC? As far as I understood, it should not be a problem as long as I don't have perfect multicollinearity since I don't mind if the standard errors get inflated. However, what about using the Likelihood-ratio test to do feature selection? I assume, I then need to look for multicollinearity?
 A: In the context of variable selection you have a serious problem with multicollinearity, far beyond the inflated standard errors that can be seen in  linear models that aren't developed via variable selection. A stepwise procedure will tend to choose the predictors that best match your particular data sample, and thus will tend to favor variables that, by the luck of the draw of the sample, happen to have high-magnitude coefficients for that sample but not necessarily for the underlying population.
Say that you have 2 correlated predictors. In variable selection, if both contribute equally to fitting your data sample there's a chance that you will miss both in your model if they individually are "less significant" than other non-correlated predictors (which happened to have high coefficient magnitudes for your sample) and thus were left out of your stepwise selection. Developing a predictor that combines information from both of the 2 predictors, or an approach like ridge regression, will work better. Alternatively, if one of the 2 predictors ends up substantially better at fitting your particular data sample, your estimate of its coefficient could be much higher in magnitude than its true population value, you will lose potentially useful information from its correlated predictor, and the result will not generalize well.
These are among the reasons why advice on this site is almost uniformly to avoid stepwise selection. You can test this advice by repeating your variable selection process on multiple bootstrap samples of your data. You are unlikely to have a very stable selection of "best" predictors among the bootstraps, particularly with multicollinearity.
