What is the Sharia ruling on executing a project with incorrect engineering designs despite warnings about the errors, and does the principle "what is built upon falsehood is false" apply to it?

Whoever's action was the cause of the error is the one who bears responsibility for it. If the error was due to the plans, then the office bears responsibility. But if the plans were sound and the error was yours, then you bear responsibility for it. Therefore, this issue must be presented to the judiciary or the competent authorities so that they may compel the one who erred to bear responsibility for their mistake and do justice to the innocent, based on the Prophet, peace and blessings be upon him, who said: "Were people to be given according to their claims, men would claim the blood and wealth of others. But the burden of proof is on the claimant, and the oath is on the one who denies." You can seek assistance from the letters addressed to the design office.