Lithuanian-German relations have varied very much.
Germany had occupied Lithuania during the World War 2 (1941-1944) and taken thousands to forced labor in Germany. Ironically, after a couple of years, the same German cities became the main refuge for Lithuanians as Lithuania was occupied by an even more brutal force: the Soviet Union. ~100 000 fled westwards as they knew they would be in the Soviet killing lists for their ethnicity, religiousness and/or the past (e.g. having had an intellectual job or participated in Boy Scouts movement). ~65 000 of them ended up in West German refugee camps (other sources claims ~200 000), established by Western Allies. There were 113 of them, aimed to maintain a quality of life for refugees not worse than that of ethnic Germans.
However, the life of those who have just lost their homeland was still hard and their future seemed unknown. Yet even under these conditions Lithuanians swiftly organized. By 1948 there were 158 Lithuanian schools in Germany, of them 26 colleges and even a Baltic University. All of this has been temporary, in refugee camp conditions and so did not remain. ~1952 USA, Canada, Australia and other countries agreed to accept Lithuanian refugees and so most of them departed.
However, February 16th gymnasium remains of that era (Lorscher Str. 1, Lampertheim-Hüttenfeld). It is a Catholic Lithuanian high school located in Renhof castle surrounded by a 5 ha Romuva park (named after Lithuanian pagan shrines). In 1953 when other Lithuanian schools in Germany were swiftly losing pupils to emigration and closing down this place has been acquired by Lithuanian priests in order to open a school for those Lithuanian families that remained in Germany. The language of instruction is German (as required b German law) but the connection to Lituanity remains strong. As the school serves a community that is spread all across Germany most pupils live in local dormitories, constructed in 1972 and 1987. Currently, the schools serve well beyond the German borders, having attracted Lithuanians from South America and after 1990 even from then-liberated Lithuania itself.
Prior to the 19th century, there was no single Germany - its place used to be taken by many small statelets each with its own royal family. One of them, Saxony, played an important role in Lithuanian history by effectively creating a union with Poland-Lithuania in the 17th century. Saxon kings August II the Strong and Augustus III also served as the Grand Dukes of Lithuania.
To this day Saxonian cities (e.g. Seftenberg, Sebnitz, Uebigau) are adorned by elaborate mileposts of the era, each of them covered with a coat of arms of Lithuania-Poland-Saxony. This coat of arms includes two Lithuanian coats of arms (Vytis). Such mileposts were first erected in 1721 to mark the distances from market square or city/town gates to the neighboring cities/towns. ~200 out of ~300 remains.
The Catholic church of Dresden (capital of Saxony) has the heart of August II the Strong interred (the body is in Wawel Cathedral, Krakow). A Vytis may also be seen near the tomb.