He was born in Edinburgh, the son of architect Robert Burn, and educated at the Royal High School. After training with the architect of the British Museum, Sir Robert Smirke, he returned to Edinburgh in 1812. Here he established a practice from the family builders' yard. In 1841, he took on a pupil, David Bryce, with whom he later went into partnership. From 1844 he worked in London. Burn was a versatile architect who was happy to turn his hand to a variety of styles. He designed many Scottish churches, and some important public buildings, including:
He also designed or remodelled as many as 600 country houses, including:
David Bryce went on to perfect the Scottish Baronial style of architecture.