This is almost exactly what occurred in the landmark case of Mapp v. Ohio, 367 U.S. 643 (1961). Police were searching for Virgil Ogletree on a tip he was at Mapp's apartment. They broke in, claiming to have a warrant which was never produced in court, and quite probably was only a paper never signed by a judge. They found and arrested Ogletree. They also found (according to the Wikipedia article) "a small number of pornographic books and pictures, which Mapp stated a previous tenant had left behind". Some months later Mapp was prosecuted for possession of pornography" based on this evidence. The Supreme Court overturned Mapp's conviction based on the unlawful nature of the search.
In the Mapp opinion, the Court wrote:
Since the Fourth Amendment's right of privacy has been declared enforceable against the States through the Due Process Clause of the Fourteenth, it is enforceable against them by the same sanction of exclusion as is used against the Federal Government. Were it otherwise, then, just as without the Weeks rule the assurance against unreasonable federal searches and seizures would be "a form of words," valueless and undeserving of mention in a perpetual charter of inestimable human liberties, so too, without that rule, the freedom from state invasions of privacy would be so ephemeral and so neatly severed from its conceptual nexus with the freedom from all brutish means of coercing evidence as not to merit this Court's high regard as a freedom "implicit in the concept of ordered liberty."
... This Court has not hesitated to enforce as strictly against the States as it does against the Federal Government the rights of free speech and of a free press, the rights to notice and to a fair, public trial, including, as it does, the right not to be convicted by use of a coerced confession, however logically relevant it be, and without regard to its reliability. Rogers v. Richmond, 365 U. S. 534 (1961). And nothing could be more certain than that, when a coerced confession is involved, "the relevant rules of evidence" are overridden without regard to "the incidence of such conduct by the police," slight or frequent. Why should not the same rule apply to what is tantamount to coerced testimony by way of unconstitutional seizure of goods, papers, effects, documents, etc.? ...
Moreover, our holding that the exclusionary rule is an essential part of both the Fourth and Fourteenth Amendments is not only the logical dictate of prior cases, but it also makes very good sense. There is no war between the Constitution and common sense. Presently, a federal prosecutor may make no use of evidence illegally seized, but a State's attorney across the street may, although he supposedly is operating under the enforceable prohibitions of the same Amendment. Thus, the State, by admitting evidence unlawfully seized, serves to encourage disobedience to the Federal Constitution which it is bound to uphold. Moreover, as was said in Elkins, "[t]he very essence of a healthy federalism depends upon the avoidance of needless conflict between state and federal courts." 364 U.S. at 221. Such a conflict, hereafter needless, arose this very Term in Wilson v. Schnettler, 365 U. S. 381 (1961), in which, and in spite of the promise made by Rea, we gave full recognition to our practice in this regard by refusing to restrain a federal officer from testifying in a state court as to evidence unconstitutionally seized by him in the performance of his duties.
Evidence seized in an unlawful search may not be admitted into evidence in trials in the US, either at the state or federal levels. This applies whether the evidence was what was being sought, or was an unrelated discovery. Also, information found in such a search may not be used to uncover evidence elsewhere and later, and if it is, such evidence is also inadmissible. This is the "fruit of the poison tree" doctrine.
There are limited exceptions. If the same evidence would inevitably have been found by a lawful method, it may be admissible. If it was in fact independently revealed by a lawful source it may also be admissible. But in general, evidence seized in an unlawful search, including one where there was no probable cause, is not admissible in US courts.
Note that other evidence, found by lawful means, remains valid even if an unlawful search occurred.