Don't forget to call him tomorrow.

Literal

Tomorrow him [to-に] phone [object-を] do [nominalizer-の] [object-を] forget-not [so-that-よう].

の nominalizes 電話をする into the object of 忘れる. ~ないよう (short for ~ないように) expresses purpose or admonition — 'so that you don't forget.' The truncated よう without に is slightly literary.