My annotated edition of Moby-Dick, or, The Whale (Bobbs-Merrill, 1964), edited by Charles Feidelson, Jr., contains this very brief explanatory note about "asses' ears":
Likewise, Kathleen Kier, A Melville Encyclopedia: A–Loom (1990) has this:
ASSES' EARS Handles so shaped. M[oby] D[ick, chapter] 15
And Clarence Babcock & David Maurer, Some Expressions from Herman Melville (1959) has this:
asses' ears: n. Ear-shaped lugs or handles (as explained by Willard Thorp in his annotated ed. of Moby-Dick). — 1851 Moby-Dick I, 80
Thorp's edition of Moby-Dick was published by Oxford University Press in 1947. Unfortunately, I haven't been able to find a searchable copy of it online, so I can't reproduce his explanation here.
Mary Reichardt's edition of Moby-Dick (Ignatius Press, 2011) goes into somewhat greater detail than the other sources cited here:
asses' ears: iron supports that look like the ears of donkeys.
Interestingly, a certain Greek bottle is described in W. Harris, Homiletical Commentary on the Books of Samuel (1884) as having similarly shaped handles:
Reland adduces a great number of quotations from Greek writers, showing that the ancients used a bottle with two long handles, which, from their resemblance to asses' ears, were called (ονοι) asses ; and the Greek poet Sosibus says of one of his heroes, "He ate three times in the space of a single day three great asses of bread, which Casaubon understood to signify the lading of three asses, whereas the true meaning is the contents of three vases or jars called asses." (Jamieson).