You can obtain several datasets from ICWSM conference (http://www.icwsm.org/2015/datasets/datasets/) as well as from SNAP Platform by Stanford University (http://snap.stanford.edu/data/twitter7.html).
The library of congress appears to hold a copy of every tweet: http://blogs.loc.gov/loc/2010/04/how-tweet-it-is-library-acquires-entire-twitter-archive/
Though, I don't know if they make the archive available as a dataset.
It contains 1,600,000 tweets extracted using the twitter api . The tweets have been annotated (0 = negative, 2 = neutral, 4 = positive) and they can be used to detect sentiment .
There is also this available-on-request dataset of Twitter true and fake profiles from the Italian CNR. It also contains tweets from the scraped profiles: http://mib.projects.iit.cnr.it/dataset.html
Shahriar Mohammadi you can access realtime Twitter data via the Search API using DiscoverText.com. The trial is free for three days and students get a heavy discount.
Pls note that most Tweeter datasets are not publicly available because of the privacy! Most researchers who collected and extracted tweets from Twitter are not allowed to publicly distribute the original tweets, such as FA Cup Final, Super Tuesday for US Elections and US Elections datasets:(http://www.socialsensor.eu/results/datasets/72-twitter-tdt-dataset)
So, If you are going to develop a research/commercial project on Twitter, you have two options:
1. Construct your own dataset through Twitter Streaming API.
2. Use some limited Twitter datasets that are publicly available under strict regulations such as:
If you need a Twitter data of a particular hashtag or search term then either purchase it from Twitter or from third party service providers for Twitter dataset like https://www.trackmyhashtag.com/historical-twitter-data .
For random Twitter dataset, look for the following links: https://github.com/shaypal5/awesome-twitter-data http://academictorrents.com/details/d8b3a315172c8d804528762f37fa67db14577cdb https://github.com/zfz/twitter_corpus
If this database can be reduced to public tweets than include certain keywords of your interest, a social media monitoring tool should be a good solution. It collects all the tweets, that include monitored keywords, automatically in one place and provides you with a self-updating database. You can check how the tool, I am a community specialist for, works, which is Brand24 (https://brand24.com/) or search for other available solutions that also monitor and collect Twitter's public data.