Dr. Lanyon is described as a "hearty, healthy, dapper, red-faced gentleman, with a shock of hair prematurely white, and a boisterous and undecided manner." He is a gentleman of equal social stature to Mr. Utterson and Dr. Jekyll – in fact, the three of them have been friends since childhood. He and Dr. Jekyll, however, had a massive scientific argument about ten years before the events of the book take place, and their friendship is more civil than friendly. At one point Dr. Jekyll refers to Mr. Lanyon as "a hide-bound pendant" – from which we can deduce that he’s more rational, scientific, and systematic than his former friend.
So what’s up with his death? Well, having just said all that about Lanyon being a man of science and a firm believer in logic, you can imagine it would be pretty shocking for such a gentleman to watch his best friend undergo a supernatural switcheroo from decent and God-fearing to evil-oozing. Such a sight would be difficult for Lanyon to believe. So, rather than believe it, he dies.