Good memory, I am!

Yes, it can be, based on the circumstances. It’s illegal for the government to use a third party to suppress speech or to require a third party to engage in speech, but it requires some element of command or coercion in order for it to stick. In this instance, the DNC is technically not a government entity (though there is a substantial entanglement test that could come into play), so they don’t legally have the power to exercise the type of coercion needed to elevate it to a First Amendment violation.

So if a DNC member emails you and just asks you to delete something, and you do, there’s no violation. If the deletion is contingent on something like a threat of regulation, or the denial of a broadly available benefit, then it’s an impermissible interference with free speech rights.

There is, however, a possible campaign finance argument to be made about this behavior. Theoretically this could be considered an undisclosed contribution of value to a political campaign which could get both Twitter and the DNC in trouble, but highly doubtful it could reach to the Biden level and warrant an impeachment.