Green’s Dictionary of Slang

no burner of navigable rivers phr.

[play on colloq. phr. set the Thames on fire, to accomplish a noteworthy feat ]

a phr. used to describe an unexceptional person, one who will make no mark on the world.

[UK]Grose Classical Dict. of the Vulgar Tongue ms. additions n.p.: Burner. He is no Burner of Navigable Rivers; i.e. no Conjuror, or rather he is a fool.
[UK]Grose Classical Dict. of the Vulgar Tongue (2nd, 3rd edn) n.p.: He is no burner of navigable rivers; i.e. he is no conjuror, or man of extraordinary abilities; or rather, he is, but a simple fellow.
[UK]Lex. Balatronicum.
[UK]Egan Grose’s Classical Dict. of the Vulgar Tongue.