The last German Emperor, Wilhelm II, was practically exiled to the Netherlands after WWI, and he spent the rest of his life there. Wilhelm II died in mid-1941 and was given a small military funeral by a small detachment of the German Army which had essentially been guarding him (against the will of Hitler, mind you, who became rather pissy after hearing of it).
Wilhelm II played practically no role in WWII. He was, however, a bit kooky because when Germany was successful during the early stages of the war he took credit for it and thought that he would somehow get back the throne. On one occasion, after Nazi victory in the Netherlands, he wrote a letter to Hitler that read: "Congratulations you have won using my army!" In another occasion he wrote to Hitler congratulating him for the victory and hoping that Hitler would restore the monarchy.
His hopes to restore the monarchy, by the way, was absolute lunacy - there is a, perhaps, anachronistic story that upon reading this letter Hitler had turned to one of his aides and said something along the lines of, "What an idiot!" Hitler, and Nazism as an ideology, was staunchly anti-Monarchy. In fact, he, as well as Nazi ideology as a whole, put a great deal of the blame for losing WWI on the royal family and the German monarchy.
Kaiser Wilhelm II refused an offer by Churchill to flee to Great Britain and escape the Nazis because, in part, he didn't like the British as he felt them too liberal and indulgent in evils, as well as that Britain was ruled and manipulated by evil Jews. So he stayed in the occupied Netherlands where Hitler had a detachment of Wehrmacht soldiers serve as essentially bodyguards to both protect him and keep people. Hitler, of course, didn't like the ex-Kaiser, but couldn't really justify or get away with killing a huge symbol of the German Empire and the Imperial German era, so it was more politically expedient to pretty much just isolate him by guarding him where he lived. He died about a year later in mid-1941, and, well, I've covered this already :P
Beyond this, the members of the royal family didn't do much. I know that a few of them fought in the war (one of Wilhelm's sons and eight of his grandsons) for the Germans. He even bragged about, in one occasion in 1939, after the Nazi victory against Poland, he wrote to Hitler (through his, the Wilhelm II's, adjutant) about the House of Hozenhollern had remained loyal to Germany and the aforementioned Prussian princes were fighting for Germany at the front, and some of them would die in it (such as his grandson, prince Oskar). Others would be involved in Nazi Germany in differing ways (e.g. Wilhelm's son, prince Oskar, the father of the aforementioned prince Oskar, was in the SA).
Perhaps the most interesting of Wilhelm's issue during this time period is his son, Crown Prince Wilhelm, who stayed in Germany. Late in the war when plans to replace Hitler were being floated by certain wings of the military and diplomatic service, he was approached by them with the proposal to replace Hitler and bring back the monarchy, but he turned them down. After the failed assassination attempt in 1944 he ended up under watch by the Gestapo.