Demur is primarily a verb meaning (1) to object, or (2) to hesitate because of doubt. Some dictionaries also list it as a noun referring to the act of demurring, but the word usually gives way to demurral for this sense. Demure means (1) modest and reserved, or (2) affectedly shy. It is only an adjective.
Demur becomes demurred, demurring, and demurs. Demurely is demure‘s corresponding adverb. The words are not quite homophones; demure is pronounced de-MYUUR, and demur is pronounced di-MUR.
They have local addresses, but if you call and ask to visit, they demur. [New York Times]
Every time our chat even hovered near the boundaries of this man’s fame, he demurred and flapped his hands to indicate he didn’t want to talk on those terms. [Independent]
She recorded his weaselly demurral: “We’re not trying to be anti-Semitic. We’re trying to be pro-human rights.” [City Journal]
Canadians are known as a demure lot, none too comfortable with blowing their own horns. [Vancouver Sun]
She held her hands demurely in her lap, a still presence with an air of vulnerability. [Los Angeles Times]
Her demure pose with her arms held across her chest helped preserve her modesty. [Daily Mail]