Does it make sense to use a tfidf matrix for a model which expects to see new text?
The problem is not really "new text", since by definition any classification model for text is meant to be applied to some new text. The problem is out of vocabulary words (OOV): the model will not be able to represent words that it didn't see in the training data.

The most simple way (and probably the most standard way) to deal with OOV in the test data is to completely remove them before representing the text as features.

Naturally OOV words can be a serious problem, especially in data such as Twitter where vocabulary evolves fast. But this issue is not related to using TF-IDF or not: any model trained at a certain point in time can only take into account the vocabulary in the training data, it cannot guess how future words are going to behave with respect to the class. The only solution for that is to use some form of re-training, for instance semi-supervised learning.