Note: As of February 9, 2023, Twitter will no longer support free access to the v1 or v2 API endpoints. Details about a paid basic tier are forthcoming and there will hopefully be clarification about how these changes will effect the academic researcher track.
On January 26, 2021, Twitter announced the "Academic Research product track" that allows academic researchers to access historical tweets (as well as other features; see the link for the full details).
Below are links to a few tweet ID datasets that are freely available online, though some strategic Internet searching may yield other options. In order to acquire tweets from the Twitter API based on these saved tweet IDs, you'll need to make use of a tweet hydrator, such as Documenting the Now's Hydrator.
Learn more about Twitter data structure and preservation with the Data Curation Network's Twitter Data Primer.