Learn Icelandic: Eyrun eru viðkvæm líffæri. - The ears are sensitive organs.

Eyra is a weak neuter noun. In the nominative plural, the ending changes from -a to -u, giving us eyru. To make it definite ("the ears"), we attach the plural neuter article -in. Since eyru already ends in a vowel, the -i- in the article drops out, leaving us with eyrun.

Do the other weak neuter body parts follow this exact same plural pattern?

Yes, this small closed set of words (auga, eyra, hjarta, lunga) all take an -u in the nominative plural. This gives us augu (eyes), eyru (ears), hjörtu (hearts), and lungu (lungs). Notice that hjarta undergoes a U-shift (U-mutation) to become hjörtu because of the added -u.

Why doesn't the adjective viðkvæm have a plural ending like -ir or -ar?

Adjectives must match the gender, number, and case of the noun they describe. Here, both eyrun and líffæri are neuter, plural, and nominative. The neuter plural nominative form of most adjectives has no suffix (it is just the bare stem), making viðkvæm the correct plural form of viðkvæmur.