Law and the Multiverse uses comics and movies to explain the law; today they turn their hands to the evidence that Batman provides to Commissioner Gordon, and how district attorneys like Harvey Dent would be constrained in using that evidence to prosecute the crooks that Batman helped catch.
Law and Superheroes, the Law and the Multiverse book, is a fantastic legal text, explaining this and many other subjects in depth. It's like a first year lawschool survey-course where all the cases and references come from your favorite superhero stories!
Now, the best (most interesting) for last: search warrants. In the ordinary case, police need a search warrant based on probable cause to search for and seize evidence. Although inapplicable to the bank example above, we can imagine a situation in which Batman infiltrates a warehouse and identifies a large hidden cache of Penguin's weapons. Batman alerts Commissioner Gordon to the location of the weapons. Before the GCPD can search the warehouse and seize the weapons it must first obtain a search warrant from a magistrate judge. In the usual case, a police officer who has witnessed what she suspects to be evidence of crime appears applies to a magistrate for a warrant based on what she witnessed. In the present case, Commissioner Gordon has not personally witnessed the weapons cache. Being an honest cop, the Commissioner will not lie under oath to obtain a warrant. Luckily, under the Constitution the police, and our Commissioner, can obtain a warrant based on a tip from a reliable informant!
A search warrant based on an informant's tip (hearsay) requires that the totality of the circumstances indicate to the magistrate judge that there is a fair probability that contraband or evidence of crime will be found at a particular place.  The totality of the circumstances test takes into account the truthfulness and accuracy of the informant, as well as the basis of the informant's knowledge. Since this test is not rigid, probable cause could be found on facts provided by an informant either because the informant has been reliable in the past or because evidence of the informant's basis of knowledge of those facts is strong.
Lacking independent police obtained evidence of crime (i.e. Harvey Bullock sees the cache of weapons himself), Gordon can appear before the magistrate judge (or provide an affidavit) to testify that: (1) he received a tip that there is a large cache of weapons in a warehouse in Gotham; (2) the warehouse is owned by Penguin; (3) the tip was provided by an informant with the alias "Matches," (a named informant because Commissioner Gordon regularly relies on tips received from Batman); (4) "Matches" has supplied truthful and accurate tips in the past (meaning he has given tips in the past that resulted in the GCPD finding what "Matches" said it would find); and, (5) "Matches" personally gained access to the warehouse and saw the weapons cache. The more specific Commissioner Gordon can describe the weapons cache and the warehouse, the more likely the warrant issued on Batman's tip is likely to withstand its subsequent challenge by Penguin after he is arrested. Thus, the tip is more reliable when Batman takes an inventory of the weapons and informs Gordon that Penguin has 20 cases of fully automatic Uzi's and 10 cases of RPG's. Likewise, the more the GCPD uses "Matches" as an informant who leads it to evidence of crime, the more reliable "Matches" becomes as an informant!
Batman and the Constitution: How can the Gotham D.A. convict criminals captured by Batman?
[Kevin Lelonek/Law and the Multiverse]