How much can I quote a research article before I infringe on copyright?
If this is a legal question, the answer is "it depends". On jurisdiction, for one. And you can look into the 4-factor test of Fair Use if you live in the US. The "substantial portion" question is going to be most relevant here (are you citing so much that there's no point in reading the original paper any more?), and maybe the amount of commentary you surround your quotes with. This kind of use is generally only determined to be infringing or non-infringing on a case-by-case basis by the court, so there is no true answer to your question short of getting sued to determine it.

If it's just a practical question, then I don't think anyone is currently pursuing DMCA claims for quoting portions of papers, and if you make it clear who wrote the text, it's also not plagiarism (which you haven't asked about but noting it just in case).