1930s, from Dempster-Dumpster trash-hauling mechanism, patented by Dempster Brothers and probably named from dump (v.) with the surname in mind. Dumpster diving attested from 1979. Dumpster fire, in figurative reference to a situation that is calamitous, foul, and unfixable (and possibly not worth the trouble of attempting to fix) or a person perceived as a walking cascade of failures and bad decisions, emerged into popularity in 2015, in the run-up to the 2016 U.S. presidential election.
The surname (late 13c.) is a fem. form (but, like Baxter, probably used also of men) of Deemer, a North of England and Manx term for "a judge;" see deem (v.).