This is not a completely accurate statement. Appropriations are done on an annual basis. So Congress has the opportunity each year to choose the funding level for any department or program. For example, it might take a separate piece of legislation to repeal Obamacare, but Congress could legally refuse to fund it, to the point where it could not operate. Of course, they would have to be able to pass it through both houses of Congress and overcome a Presidential veto. But the fact that Obamacare was already law would not force them to appropriate funds for it - otherwise, you could never stop funding any program ever passed.
You may be thinking back to the "government shutdown" when this was the Dem talking point. Obama kept getting on TV talking about the debt ceiling, crying that the GOP was refusing to pay for bills already incurred. Of course, that was a lie. Not raising the debt ceiling would have forced the government to stop incurring new obligations above what could be paid with the discretionary income left after paying for existing obligations and mandatory spending (debt service, SS, etc). But of course the MSM parroted that lie and the gullible public bought it.
If funding must be approved in each year for all past and current bills and laws, then you are entirely correct.
But then does not that essentially give the House veto power over existing laws? In other words the House could kill/veto any existing law by simply denying the funds to execute the law.
Do you have any examples of which laws in the past were subject to such denial of funding?