Did Japan have a military presence outside of the Pacific Theatre? Certainly. Now, depending on one's definition of "Pacific Theatre," the size of this presence can vary. In America it is common to refer to the entire war against Japan as being the Pacific War or taking place in the Pacific Theatre - in this case, the term is really just being used as a catchall phrase. However, there were definitely major Japanese operations that took place well inland in Continental Asia, and there were also important Japanese operations in the Indian Ocean. From a technical standpoint, it is hard to consider those a part of the Pacific Theatre, which is why some people prefer the phrase "Asia-Pacific War" to describe the fight against Japan. The Japanese themselves called it the Greater East Asia War, which, while being a fairly jingoistic title given the context of the day (it is a reference to the Greater East Asia Co-Prosperity Sphere), is a name that reflects the scope of the conflict.
A major part of the Greater East Asia War was the Chinese front. Japan's war with China - known as the Second Sino-Japanese War - actually began in 1937, two years before WWII, and continued on throughout the Second World War. The conflict in China is perhaps the most widely underrepresented theatre of the entire war in proportion to its scale. There were more Japanese servicemen in China than anywhere else in the Empire for most of the war, and more Japanese servicemen lost their lives in China than in any other single location. The Japanese suffered almost as many losses against the Chinese as they did fighting the Americans. Even near the end of the war, when invasion of the Japanese home islands seemed eminent, there were still 900,000 Japanese soldiers in China - and these soldiers were of a higher quality than many of the ones who were tasked with defending Japan herself.
The Japanese also conducted major operations against the Allies in South-East Asia. From the British, they conquered most of Malaya by January 1942 and took Singapore in February. They had taken French Indochina from Vichy France back in 1940. Burma would see continuous fighting for much of the war, between Japanese forces on one side and British, American, and Chinese forces on the other. The Japanese Navy conducted temporary raids into the Indian Ocean in order to reduce the Royal Navy's presence in the region and sink British merchant vessels. The American Army used the term China-Burma-India Theatre to describe operations in this area.
The Japanese Army also maintained a sizeable presence in Manchuria (then the puppet state of Manchukuo) throughout the war - the so-called "Kwantung Army" - which was defeated quite thoroughly by the Soviets when they invaded Manchuria in August 1945.
Imperial Japanese forces did not take any part in the War in Europe. Japanese soldiers did serve in Europe - Japanese-Americans, that is. The predominantly Japanese-American 442nd Infantry Regiment is the most decorated unit of its kind in American history.
While the Japanese did not fight in Europe, there were delegations and Japanese submarines operating in the Atlantic. The subs were part of the Axis blockade-running effort in which the Japanese would exchange scarce raw materials like tungsten for German technical knowledge and plans. As such, the submarines would avoid combat or Allied shipping lanes as it put their mission at risk. These efforts were in vain as the Allies were able to sink most of the Japanese submarines and their German counterparts because of Allied intelligence intercepts.
There were also a number of Japanese military personnel active within the European theater. These were not combat troops, but high-ranking officers and technical specialists. For the most part, neither side gained much from this relationship. The German surface fleet had more or less become harbor-bound by the time of Pearl Harbor, so the IJN's expertise in surface warfare was of minimal use. The Germans gave technical support, but being a continent away meant that the home islands could not use this knowledge in a meaningful way as plans and personnel had to run the gamut of an increasingly tighter Allied naval superiority.
The one major operation in which the Japanese were consulted in a serious manner was Operation Herkules, the stillborn Italian-German invasion of Malta. IJA and IJN attaches attended Kurt Student's conferences for the invasion as they had greater experience with combined operations. Herkules was something of an exception that proves the rule that the Axis alliance tended towards self-interest.
Sources
Boyd, Carl, and Akihiko Yoshida. The Japanese Submarine Force and World War II. Annapolis: Naval Institute Press, 2002.
Krug, Hans-Joachim. Reluctant Allies: German-Japanese Naval Relations in World War II. Annapolis, Md: Naval Institute Press, 2001.