Just going over the FR1-FR16 listing:
FR1 - Waterdeep - keep if you don't have the City of Splendors boxed set; if not, the boxed set is more useful.
FR2 - Moonshae - if you play there keep; also, the Moonshaes have not really been given a full treatment elsewhere.
FR3 - Empires of the Sands - keep only if you don't have the Lands of Intrigue boxed set (and / or the Calimshan book)
FR4 - The Magister - Not useful enough to keep;
FR5 - The Savage Frontier - I'd always keep this pne, because it's one of the best 1E books, but if you have The North boxed set, you can afford to lose it.
FR6 - Dreams of the Red Wizards - Definite Keep.
FR7 - Hall of Heroes - not useful enough to keep; most of the NPCs in there will either never be used in campaign (novel characters), or have been written up elsewhere (Elminster, and the staple NPCs)
FR8 - City System - keep it if you have it, because of the additional info on Waterdeep and the maps.
FR9 - Bloodstone Lands - These have not gotten the full treatment elsewhere, so worth keeping, but of you don't venture there, you can afford not to.
FR10 - Old Empires - this depends on which era your playing in; if it's pre-Times of Troubles, keep; if post-Time of Troubles, the lore in Lost Empires of Faerûn is better.
FR 11 - Dwarves Deep - Keep. Very good writeup of dwarven culture, and if you have a dwarf PC or some dwarf NPCs, it's worth it.
FR12 - The Horde (or, what if Osprey Publishing made a Realms supplement) - Not really worth it from a lore point of view. If you want to play with the Hordelands, the Hordelands boxed set is better.
FR13 - Anauroch - Keep. If you want to use Anauroch as a sandbox (pun intended), this is the place to be.
FR14 - The Great Glacier - corner case; if you play there, keep it; if not, then don't.
FR15 - Gold and Glory (or, what if Osprey Publishing got to make a second Realms Supplement) - not that useful.
FR 16 - The Shining South - I think the lore in this one is outdated; I use the lore in the 3E Shining South and Serpent Kingdoms as the baseline even in pre-1372 DR games.
If you have some of the boxed sets, these are worth keeping around for the maps, and casual readin): The North, Lands of Intrigue, City of Splendors. Also nice for casual reading: any Volo's Guide, the three 2E god books (Faiths & Avatars; Powers & Pantheons; Demihuman Deities); Cloak & Dagger.
I have not played much 3E, but the books that I like for some hidden gems; Lost Empires of Faerûn, Serpent Kingdoms, Shining South