Adam Drese was born in the German state of Thuringia. He became a musician at the court of Duke Wilhelm IV of Saxe-Weimar in Weimar before being sent by the duke to Warsaw to study with Marco Scacchi. He returned to Weimar and was appointed as the duke's Kapellmeister in 1652. In 1662, after the death of Duke Wilhelm IV, the Hofkapelle was disbanded, the duke's son, duke Bernhard took Drese to Jena, where he appointed him as his Kapellmeister, secretary and also the town mayor from 1672. Despite Duke Bernhards's death in 1678, Drese remained in Jena until 1683 when he found another post as Kapellmeister, to the Count of Schwarzburg at Arnstadt. He remained in Arnstadt until his death in 1701.

It is thought that most of Drese's output has been lost. His surviving pieces include sacred and secular vocal works, as well as some instrumental Sonatas and Suites. Most notably, 3 sonatas by Drese were included in the 1662 manuscript collection Partiturbuch Ludwig. He became known in his time for the composition of several chorale melodies which used the texts from Georg Neumark's Das Poetisch-musikalische Lustwäldlein and Nicolaus Zinzendorf's Jesu, geh voran. These melodies were set by many composers of following generations as part of their own sacred works.