A contraction of do-out, as don is of do-on, doff of do-off, and dup of do-up. In Devonshire and other southern counties they still say Dout the candle and Dout the fire. In some counties extinguishers are called douters.
The dram of base Doth all the noble substance dout.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894