First, if a non-law enforcement officer acts on the direction of an LEO, the action is considered to be LEO action. We see that a lot with non-LEO authorities...teachers, hospital employees, etc... who often work near law enforcement in certain situations and LEOs used to "suggest" an action to the non-LEO that would help their case.
Second, the question here is more about the expectation of privacy of the person with respect to the item. Say, for instance, this guy had a partial Coke in his home. He invited a friend in who(upon suggestion or direction from an LEO) snatched up the can while visiting and took it to the police. Clearly, in his own home, the guy would expect privacy with respect to the can, and other items in that home. With that reasonable expectation of privacy, LEOs would need to obtain a warrant to conduct a search and obtain the can (and as described above, this would include a Non-LEO acting at the direction of an LEO). Now, guy in his office tosses the can in a trash can, even in his private office at a workplace, and he starts to abandon that expectation of privacy. It's fact dependent (does a custodian empty the can, is it a can in a public or private location, etc.) But, in general terms, if you throw something in the trash, you've given up your privacy rights on that item, and LEOs can seize it without need for a warrant. In my part of the world, drug enforcement cops will often seize the trash of a home suspected of drug activity and look to see if there's evidence of such in the trash. Occasionally, to maintain the covert nature of the investigation, they will match trash bags and swap out a bag of trash for the trash from the home, so no one notices the trash gone before the haulers come. Important point...in a private location, just because you put the trash in a bag/can doesn't abandon all privacy rights until you put the trash out. Example, if you keep your cans in your garage, or your backyard, etc., LEOs cannot come onto your property and take the trash. They have to wait until it hits the curb.
Now, as Mike points out, this isn't trash on the curb, it's in a can in a shared location. If I was going to place a bet, I"d say the outcome here is going to depend a lot on the judge. If the judge is a big privacy rights supporter, the finding will be that the trash was still private until collected by a custodian for disposal, or placed in a receptacle for collection (put in the dumpster, etc.) If the judge leans LEO's way, I'd expect a ruling that the guy abandoned his privacy rights when he tossed the can/cup in a shared trash bin. Where it goes on appeal is anyone's guess.
Oh, and as a practical point, I'd also wonder how much the nature of the crime may impact the case. In my experience, the more heinous the offense, the more likely a judge will allow the evidence to be admissible. Judges are humans, after all, and are less likely to throw out evidence in close-call situations if the offense is violent. The more violent or gruesome, the more likely the close call will go the way of law enforcement. This is only on close call cases. No brainer situations need not apply.
