Is it possible to use Tensorflow or some similar library to make a model that you can efficiently train and use at the same time.
An example/use case for this would be a chat bot that you give feedback to. Somewhat like how pets learn (i.e. replicating what they just did for a reward). Or being able to add new entries or new responses they can use.
I think what you are asking is whether a model can be trained continuously without having to retrain it from scratch each time new labelled data comes in.
Answer to that is - Online models
There are models that can be trained continuously on data without worrying about training them from scratch. As per Wikipedia definition
Online machine learning is a method of machine learning in which data becomes available in sequential order and is used to update the best predictor for future data at each step, as opposed to batch learning techniques which generate the best predictor by learning on the entire training data set at once.
Some examples of such algorithms are