Extending a neural network to classify new objects Suppose a model M classifies apples and oranges. Can M be extended to classify a third class of objects, e.g., pears, such that the new images for 'retraining' only have pears annotated and apples and oranges ignored? That is, since M already classifies apples and oranges, can the old weights be somehow preserved and let the retraining focus specifically on learning about pears?

Methods such as fine-tuning and learning without forgetting seem to require all objects in the new images annotated though.
Yes this is standard transfer learning. Using a trained model, we can freeze the first N hidden layers of a classifier, except for the last few. This will allow our previous relevant training to be retained whilst also being able learn new features and target new classes.
We will then initialize a new output layer that works our new context(i.e sigmoid, 1 node for binary classifier). Everything is now set to resume training on our new y_targets.
Take a look at this link for some more info on transfer learning.
If you want no perturbance to your past learning, I would recommend freezing all previous hidden layers and then tacking some additional ones on.