。
Please don't forget to wake me up at 7 tomorrow.
Literal
Tomorrow forget-not [request-で] me [object-を] 7-o'clock [at-に] wake-up please.
忘れないで is a negative te-form functioning as an adverbial modifier on the main request 起こしてください ('please wake me up'). The ~ないで form serves double duty in Japanese: as a standalone negative request ('don't do X') and as an adverbial ('without doing X, do Y').