Is it possible to predict sentiment of unlabelled dataset using BERT?
Bert is a pre-trained language model with objectives like masked token prediction and next sentence prediction. So, it doesn't have any setup to do sentiment analysis,but you can use the pre-trained information to do the same.
There are couple of ways to solve your problem, they are

Annotate few of your documents and fine-tune the model for sentiment analysis.
Get open-source sentiment analysis datasets which fit your requirement, train Bert on the data and use the same classifier for your purpose (But this will work only if both data distributions look the same).