IT DEPENDS ON WHAT YOU MEAN BY 'COST.'
In politics, the words are often moulded to have meanings beyond what they would mean in common, conversational contexts. Here, the word 'cost,' is being applied in a very specific way.
The Biden Administration is claiming that the program will not require any increase in the national debt. That's why the tweet you cite ends on that point, to clarify what is meant by 'cost' in that statement. It's repeated in the first quote cited by the Post
This isn't necessarily disingenuous, insofar as 'cost' can be taken to mean "a burden borne as a result of a choice."
Obviously the program's various aspects will "cost" a fair bit (hundreds of billions per year over ten years) in terms of dollars being spent to make them manifest - but in economics you can find the concept of relative costs.
Suppose you were forced to choose between A: losing $500 and B: losing $700. You can describe this by saying Option A costs $500 and Option B costs $700. That's how most people would, conversationally, understand the word 'cost.'
But to an economist or accountant, the fact of the matter is that you're out at least $500 no matter what. So really the choice is between losing another $200, or not. (This is known as throwing good money after the bad.) In that context, you can say "making the 'wrong' choice will cost you $200." This is not an incorrect way of describing it, and is similar to the way in which the Biden administration is claiming their plan will cost $0. They're basically saying the money that they're going to use to fund this is already committed, or would have been spent for other things.
The term "budget neutral" is used to mean something similar here, but they're probably avoiding that term either because they don't think it polls well or because there's some other slightly more creative accounting they're claiming (such as raising taxes on rich folks).
It's certainly not as honest as speaking in plain language, but it's not all smoke and mirrors, either.