ear-muff n.

"organ of hearing," Old English eare "ear," from Proto-Germanic *auzon (source also of Old Norse eyra, Danish øre, Old Frisian are, Old Saxon ore, Middle Dutch ore, Dutch oor, Old High German ora, German Ohr, Gothic auso), from PIE *ous- "ear" (source also of Greek aus, Latin auris, Lithuanian ausis, Old Church Slavonic ucho, Old Irish au "ear," Avestan usi "the two ears").

þe harde harte of man, þat lat in godis word atte ton ere & vt atte toþir. [sermon, c. 1250]

In music, "capability to learn and reproduce by hearing," 1520s, hence play by ear (1670s). The belief that itching or burning ears means someone is talking about you is mentioned in Pliny's "Natural History" (77 C.E.). Until at least the 1880s, even some medical men still believed piercing the ear lobes improved one's eyesight. Meaning "handle of a pitcher" is mid-15c. (but compare Old English earde "having a handle"). To be wet behind the ears "naive" is from 1902, American English. Phrase walls have ears is attested from 1610s. French orielle, Spanish oreja are from Latin auricula (Medieval Latin oricula), diminutive of auris.

1590s, "covering into which both hands may be thrust to keep them warm," from Dutch mof "a muff," shortened from Middle Dutch moffel "mitten, muff," from  French moufle "mitten," from Old French mofle "thick glove, large mitten, handcuffs" (9c.), from Medieval Latin muffula "a muff," a word of unknown origin.

The muff was introduced into France toward the close of the sixteenth century, and soon after into England. It was used by both men and women, and in the seventeenth century was often an essential part of the dress of a man of fashion; but it is now exclusively an article of female apparel. [Century Dictionary]

Meaning "vulva and pubic hair" is from 1690s; muff-diver "one who performs cunnilingus" is from 1935.