Well, regarding the OP (& I apologize if someone's already mentioned this):
IIRC, the original non-human races of dwarf, elf, and hobbit/halfling did have a strong Tolkienesque feel/style to them. Now, there are plenty of other sources for such fantasy races, but these 3 have strong roots to Tolkien's work. However, I'd also argue that to a certain degree, these core 3 races took on slightly stronger associations with Tolkien's versions of these fantasy folk as the game evolved, even after Gygax was no longer affiliated with TSR.
Hobbits/Halflings are the most blatant link (apart from Ents/Treants and Balrogs/Balors). The physical description of them was dead-on, and the later evolution of them into the 3 subraces of Hairfoot, Stout, and Fairhair mirror Tolkien's 3 varieties of hobbit: Harfoot, Stoor, and Fallohide. Halflings only stopped bearing this resemblance after the "kender infusion" they received in 3.0/3.5.
Dwarves are a bit less blatant, but still seem to derive from the Tolkien source material quite a bit. I think the main reason why is the severe lack of magical ability attributed to them. They're underground craftsmen, as per myth, but have more of a warrior ethic akin to Gimli & Thorin & Co. The hill/mountain dwarf split still kinda fits the Tolkien mold (IMHO, Dain & his forces from the Iron Hills are the AD&D hill dwarves, whereas Thorin & the dwarves of Erebor are AD&D mountain dwarves). However, AD&D starts to step away from Tolkien & expand in the dwarf varieties with Grey Dwarves/Duergar & the other dwarf subraces (IMHO, the Duergar are more akin to the malicious magical dwarves of myth, like Regin from Sigurd's tale).
Elves are in the same boat as dwarves, but (IMHO) slightly more removed from the Tolkien source material. The initial 3 varieties of AD&D elves, High, Grey, and Wood, mirror the 3 varieties of Elves in Tolkien's work, the Rivendell elves, Lothlorien elves, and the Mirkwood elves. However, other tales of elfkind have an influence--most notably, the AD&D elves are human-sized, though generally smaller than humans (rather than as tall if not taller than humans as Tolkien's elves were). The initial AD&D elves seem to mirror Tolkien's elves, though the height seems to be a slight nod to the diminutive dwarf-like or pixie-like elves of myth. However, as the dwarves, further versions of the elves depart from the Tolkien source; most notably, the drow mirror the dark elf/light elf split of Norse myth. Aquatic elves seem to be a more wholecloth creation, though the seed of the idea may be from Tolkien (mainly the name, as the Sea Elves from Tolkien, IIRC, were just seafaring elves, and not aquatic elves).
Gnomes seem to take over the role of the mythical dwarf craftsman/trickster in AD&D. IMHO, they fit the niche of the spellcasting dwarf that the Tolkienesque dwarves can't (and, I suspect, were added because of that). Three Hearts and Three Lions has a significant contribution to the gnomes (IIRC), as well as to trolls, paladins, and swanmays. I suspect that if the AD&D dwarves were more of a fusion between Tolkienesque dwarves and mythical dwarves, then gnomes wouldn't have existed as a seperate race (or would be a subrace of dwarves rather than a seperate and distinct race).
I think prior analysis of orcs pretty much covers them. However, I'd like to add that the orc/goblinoid group as a whole seems a bit more Tolkien-derived: goblins represent the Azog's goblins (from the Hobbit), the goblins of Moria, and the "small orcs" Sam and Frodo masquerade as. Orcs represent the "standard" orc of LotR that the Fellowship must deal with. Hobgoblins fit more in line with the Uruk-Hai, the sun-resistant, more soldier-like goblinoid. Bugbears, however, seem to be a late mythical addition (since, IIRC, they weren't a goblin-like type of beast to start with in D&D).
The ranger seems to be a tip of the hat to Strider/Aragorn (esp. the magical abilities of the class). However, other classes don't seem to be restricted to the Tolkien mold: fighters and magic-users/mages/wizards are commonplace enough; clerics are priest-knights; thieves/rogues hearken to the Grey Mouser and larcenous activities of Conan (not to mention the plentiful real-world influences/sources); paladins are myth & legend, though Poul Anderson's rendition seem to have a good influence as well; bards seem a bit Fflam, but derive a lot from myth; druids really derive from myth/lore; barbarians and cavaliers/knights derive from historical sources and legends (though the 3.X barbarian shifts from barbarian to berserker); monks seem to be the initial OA-class in AD&D (being more in line with Wuxia monks than European ones); assassins are a mix of history and legend (esp. legendary when magical abilities are assigned to them); illusionists seem to cover the more trickster-like deluding wizards of myth, as well as having a good dose of stage magician in them (the latter due to the Dex prereq of the initial class).
The only other Tolkienesque element in D&D is mithril/mithral. Adamant is more akin to mythical adamant than the adamant in Tolkien's setting (which was or was akin to diamond).