Predict a variable that is not in the input sequences with LSTM-Keras
I'd say that one way for it to do this is for your network to simply predict C or have C as the label
I have been seeing this again and again. Don't confuse a NN with something more than it actually is. You simply approximate the output Y given an input X by learning a function F. That is your NN.
In your case the output could very easily be C + Other_Output
Depending on what that other output is, your network could converge and have good results. It could very well not, so your question is simply at this point incomplete.You have to ask yourself some questions like:

Does C + Ohter_Output make sense for the give input.
Is there a good way for me serialize the C + Other_Output ? Like having the first K out of N output array elements describing C and the rest N-K describing Other_Output ?
Is C a multiclass problem and if so is Other_Output a different kind of problem or could potentially be turned into a multiclass of the same kind, problem that could converge along with C or make them both multilabel problem ? 

These are at least some of the questions you need to ask yourself before even choosing the architecture.
That being said, no, unless you train your network to learn about patterns between A B D and C it will not be able to predict a missing input.
Good luck,
Gabriel
Assume that I have five columns in my dataset (A,B,C,D,E) and I want to build an LSTM model by training just on A,B,D,E (i.e. I want to exclude C)
My problem is that I still want to use this model to predict C. Is it possible if I didn't train my model with this variable? How can I do that?
EDIT 1
I'm working with categorical and numerical data modeled as time series. In this specific case, C is a categorical time series (given in a one-hot representation).