Let Say:

Player A: Tribute Summon Zaborg the Thunder Monarch

Zaborgs the oppoent's f/d monster

Player B: Chains Book of Moon to the effect

Does Zaborg get his effect off, or Book of Moon prevents that from happening?

Zaborg does not need to be face up on the field (or even on the field for that matter) for it's effect to resolve properly....so yes, Zaborg would still destroy a monster if it's effect was chained to with BOM turning Zaborg face down.

Aslong as Zaborg is tribute summoned then Zaborg Gets its effect.
Unless Zaborg is stopped by

Solumn Judgemnt
Horn of Heaven
Divine Wrath
Forced Back

Or any other effect that Negates its summon.

or skill drain, you can't forget skill drain.