In the real world probable cause is not a legitimate reason to conduct a search without a warrant. Probable cause is needed to get a judge to sign a search warrant. Now if Columbo heard muffled cries for help coming from inside, say, a closet, he could search without a warrant because the belief that someone's life is in danger would probably give him probable cause, provided other probabilities don't interfere.
Are there any forms of the word probable that I failed to mention? LOL
There are a number of circumstances that legally justify a warrantless search. One of them -- I forget the legal phrase that's generally used for it -- is where the officer has cause to believe that the evidence will be destroyed or spoliated during the time it would take him to get a warrant. And unless Columbo kept a cop posted right there in the room (on what basis?), it seems pretty obvious that the body would have disappeared within minutes after Columbo went out to get his warrant.
Still, the broader point is well taken, that "Columbo" stories sometimes take liberties with various laws -- especially the laws of probability!