transfer learning with sentiment analysis? The question is how good and what are some things to keep in mind when sentiment analysis models are tested on different datasets than they are trained on.

Say the task is to perform sentiment analysis on product reviews (unlabeled datset) - to classify positive, negative or neural. Because the data is unlabelled, a model can be trained (perhaps using logistic regression or NN) on a similar labeled dataset (say movie reviews, or product reviews) and tested on the original unlabelled dataset. 

Will something like this work? Because the words of product names that occur in the unlabelled dataset will not be words that the model was exposed to during training, during test time will these words possibly throw off the model?
I can not fully answer your questions, but would like to offer a couple of my thoughts here:
1) Transfer learning for sentiment analysis can be hard given that knowledge learned from one topic may not be not broad or general enough to perform well on the target or downstream tasks. For example, I have recently trained a neural network along with Word2Vec embedding using Twitter airline customer review data and get an prediction accuracy of 77%. However, when I use the same Word2Vec and neural network to classify some general customer review data, I got an prediction accuracy of only 35%.

2) Transfer learning in natural language processing is a hot topic and many researchers are working on it recently. 2018 sees some breakthroughs in transfer learning, i.e., Google Universal Sentence Encoder, BERT algorithm etc. I can not give you a comprehensive list here since I'm also learning. I would suggest you dive into some blog articles or even the original research articles to get a better understanding.

May it helps.