Opening or throwing out mail addressed to someone else, such as a previous tenant, is a felony with possible jail time and a statute of limitations of five years. But when is this ever prosecuted? If you know this crime has occurred, who should you report it to?
In the US, the various crimes regarding mail theft, tampering are federal crimes, esp. 18 USC 1708 and 18 USC 1702. USPS has a web page about reporting mail theft here, and ultimately it would be prosecuted by the US Attorney's office. However, it is not guaranteed that the US Attorney will prosecute every case brought to his attention, instead, the case might be turned over to local authorities. Many (all?) states have their own laws covering mail theft, such as Cal. Pen 530.5(e), so it is possible that the prosecution (for the state crime) would be carried out by the state's district attorney. I do not know if there are any statistics on how frequently state and federal prosecutors decline to prosecute for throwing away or opening someone else's mail, compared to them prosecuting. The Dept. of Justice suggests accepting pleas to misdemeanors before Magistrate Judges over felony prosecutions, for less serious violations. Reporting the crime to the USPS Postal Inspector seems to be the standard first step.