Question: Is it possible to get a refund from a publisher if I bought an article that contains a major flaw (i.e.,one that invalidates the main results or the main conclusions)? Assume the article was bought through one of the main academic publication's paywall.

Answer: 
There is a case where I think you might have a very strong moral claim even if not a legal one: if the paper is retracted, there is no question that you ought to be able to obtain a refund.

**Addendum** One argument contrariwise: The last science needs is any further disincentive for publishers to retract papers. Maybe better for the enterprise as a whole to have a few readers get cheated than to have fraudulent and otherwise untrustworthy papers remain in the literature.
To be successful in a request for a refund, either the publisher would have to be remarkably generous, or they would need to conclude that it is in their best interest. I'm skeptical that appeal to threatened "public hue and cry" would persuade them, since academic publishers are generally immune to negative publicity. A potentially more powerful motivating force would be legal pressure, for which I see two bases. One would be fraud, where the publisher knowingly represented falsehoods as truth. It would be challenging to establish fraud. The second is via warranty of merchantability (in the US enshrined in law via the Uniform Commercial Code article 2). In some cases, e.g. Taylor & Francis (following the lead of every known software producer), publications are offered as-is, in which case you have no option. Any product without an as-is disclaimer carries a warranty, to the effect that if the seller knows (or should know) that the item is used for a particular purpose, then the item is fit for that purpose. This is why one can sue a company for negligence, if they sell something as fit for a purpose when it is not.

In order to make any traction with a threat for violating the implied warranty of merchantability, you would have almost certainly have to enlist the aid of lawyers willing to help you pursue the matter. The two main things that you would have to objectively establish is that the article has a known purpose, and that the article actually is not fit for that purpose. I don't believe that you would have to establish that the publisher was aware of the defect (I'd like to be more assertive about that, but that's what lawyers do). 

If you just email them asking for a refund, they will probably say that they don't make any guarantees. However, if your attorney uses suitable language, they would probably reply that they have a stated as-is policy (you're out of luck), or, that there is no way they could know of the mentioned purpose, or that the article is in fact fit for that purpose. Since "merchantability" isn't defined under the law, courts would generally compare the item in question against comparable products. The reason is that no product can be absolutely flawless, and a claimed based on merchantability would have to show that the product was egregiously defective, not just less than satisfactory.

That all said, you did not actually purchase "the article", you purchased a license to copy the article in a particular manner, for which reason it is not clear that UCC article 2 is applicable.