The WikiQA corpus is a new publicly available set of question and sentence pairs, collected and annotated for research on open-domain question answering.
Usually on TREC (Text REtrieval Conference) there is a QA task which has some kind of datasets associated with it. Most of the datasets are focusing on factoid QA task but the one in 2015 is a kind of live QA. The task was to answer questions on Yahoo Answers.
Manually-generated factoid question/answer pairs with difficulty ratings from Wikipedia articles. Dataset includes articles, questions, and answers.
There are some manually curated QA datasets from Yahoo Answers from Yahoo.
You also can download the Stack Overflow questions and answers. It's a domain specific but huge dataset.

