How to force a classifier to forget some wrong cases without re-training the whole system?
First of all, what kind of classifier? I assume, support vector machine?
Since SVM builds its decision hyperplane as a linear combination only of a subset of training samples (support vectors), the rest of the training set does not affect it at all. If you wrong cases belong to the 'rest part', not to the support vectors, you do not have to worry about them. How to check it depends on the particular tool you use for the training.
Otherwise, I am afraid, SVM should be retrained.
If the classifier is an SVM, "unlearning" of single samples can be done using the methodology described in
Cauwenberghs, G., and Poggio, T. "Incremental and decremental support vector machine learning." NIPS 2001
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