He was born at Düsseldorf and brought up to the profession of art by his father, the engraver Karl Ernst Christoph Hess (1755 – 1828). Karl Hess had already acquired a name when in 1806 the elector of Bavaria, having been raised to a kingship by Napoleon, transferred the Düsseldorf academy and gallery to Munich. Karl Hess accompanied the academy to its new home, and there continued the education of his children.

In time Heinrich Hess became sufficiently master of his art to attract the attention of King Maximilian. He was sent with a stipend to Rome, where a copy which he made of Raphael's Parnassus, and the study of great examples of monumental design, probably caused him to become a painter of ecclesiastical subjects on a large scale. In 1828 he was made professor of painting and director of all the art collections at Munich.

He decorated the Glyptothek and the Allerheiligen-Hofkirche at Munich with frescoes; and his cartoons were selected for glass windows in the cathedrals of Cologne and Regensburg. Then came the cycle of frescoes in the St. Boniface's Abbey, Munich, and the monumental picture of the Virgin and Child enthroned between the four doctors, and receiving the homage of the four patrons of the Munich churches (now in the Pinakothek).

Before testing his strength as a composer Heinrich Hess tried genre, an example of which is the Pilgrims entering Rome, now in the Munich gallery. He also executed portraits, and twice had sittings from Thorwaldsen (Pinakothek and Schack collections). His last work, the “Lord's Supper,” was found unfinished in his atelier after his death in 1863.

The 1911 Encyclopædia Britannica assesses his work as follows:

His fame rests on the frescoes representing scenes from the Old and New Testaments in the Allerheiligencapelle, and the episodes from the life of St Boniface and other German apostles in the basilica of Munich. Here he holds rank second to none but Overbeck in monumental painting, being always true to nature though mindful of the traditions of Christian art, earnest and simple in feeling, yet lifelike and powerful in expression. Through him and his pupils the sentiment of religious art was preserved and extended in the Munich school.