Can a Logsitic Regression model continue making predictions after removing predictions from the data set?
The model is able to predict churn on any sample. The issue is that increasing the „customer life“ by 30 days does not lead to any change in the prediction (based on what the model has learned). 

I think you can check two things:

A) More of a fundamental thing, try to improve the model, e.g. by applying regularization to get a better fit on you features. GLMNET is the thing to go for in this case: https://web.stanford.edu/~hastie/glmnet_python/

B) Check how „sensitive“ your predictions are to changes in features. You could plot or look at predictions in case you increase „customer life“ by say 10, 20, 30, ..., 100 or so days. Remember that you basically predict the probability of churn. So it is possible that (some) probabilities at 30 days of customer life are just below 50%. If you gradually look at increases of customer life, you get a good idea of how churn changes in X. This is kind of a marginal effect (dy/dX).