You don't want to demean your dependent (response) binary variable. So you almost always want to keep beta0 to control for any imbalance in your dependent var.

I meant demeaning the independent variables. My understanding is that the beta_0 will have the meaning the curiousgal attach it only if you demean your independent variables.

I see. But I think after demeaning X, beta0 will just have a special meaning... log odds of the average case. Nothing more.