What kind of court cases can the US President not interfere with?

The US President can't interfere with any court cases without breaching the separation of powers outlined in the Constitution. The Judicial branch has sole authority over adjudicating cases.

Perhaps you're thinking of Senate impeachment (or removal) trials. The Legislative branch (Congress) has exclusive authority over impeachment proceedings, which is a process by which government officials may be removed from office. The House of Representatives files articles of impeachment (formal complaints), then votes to determine whether there is sufficient proof to send the case to trial in the Senate. The Senate conducts a trial, then votes to decide whether the official should be removed from office.