I am working on a D&D campaign where the perceived villain, who you are set forth to stop's home was taken over by invaders and they are soul survivor. It is the people whom took over this area that are trying to have them killed, and are actually the true villains of the story.

So my question is what racial groups your think receive a lot of hatred and racism? I'm aiming to make this an emotional story that has player characters reflecting on right and wrong and considering the weight of their actions.

I plan for this perceived villain to be a wizard who is trying to get vengeance for their slain people. 

The undead, that is if you count them as race.

From what I've read, Orcs, Goblins and Kobolds. As mentioned above, the undead would inherently be a contender, but they're generally not found in significant populations and since they're generally considered to be mindless/automatons in real life, is you might find it harder to evoke some sympathy from the players. The above three are races that you could play more sympathetically but, in my experience, have a natural enmity with humans, which would suit your purposes. 

I guess it really depends on the world the game is set in, is it set in any of the existing D&D worlds or is it homebrew? I mean Orcs and Goblinoids such as bugbears and hobgoblins usually get quite a bit of hate but then again it is kind of natural as often tribes that contain these races don't really get along with the rest of the world's population, in Forgotten Realms the Drow/Dark Elves get quite a bit of hate on the surface but again this is due to their reputation and their societies in the Underdark are generally quite cruel and evil. Tieflings can generally be considered more sympathetic as they have a bad reputation due to their heritage and superstition surrounding them. In Eberron Changelings are often mistrusted and mistreated due to their ability to change their appearance which causes the other races to question their intent.

But it really depends on the setting, there are plenty of settings where Elves are the subject of racism and prejudice as depicted in the Witcher or Dragon Age. If this is your own homebrew setting you can really pick anything, you could make Halflings the subject of persecution by the human populace who once enslaved them because of their lack of military might and because they could, or it could be a world where humans are the minority and persecuted by some other race that has taken control, really up to you.

I think that this will need to be tailored to the setting, as if there is a region where you decide that the population hates Tieflings, for example, then anyone who plays a tiefling will have a bad time.

Your best bet would be something that is not typically a player race. If you ban the monstrous races, then goblins, orcs, hobgoblins, bugbears, and minotaurs are all strong contenders.

As others have stated, this is a Setting question. I like breaking old tropes sometimes so in my games over the years I've had peaceful and even helpful tribes of Giants, Orcs, Ogres, and even one benevolent Undead ruler of a small kingdom. I've also had evil Elves that weren't Drow, twisted Dwarves, and my own homebrew creations.

I'd like to see a game where there was no racial animosity but that means a huge slice out of a big RP pie because there's be less political intrigue, faction fighting, etc.

Depends on the setting. In Faerun for example, Drow are widely distrusted/feared, whereas in Eberron and Ravenloft they're not treated any differently than anyone else.

There are also races/species that are largely specific to a given setting, and the attitude towards them in bound up in that setting's history. Warforged for example in Eberron are largely seen as living reminders of the war, and so they're viewed dimly. Kender are pretty specific to Krynn and attitudes to them are pretty polarized. Yuan-Ti are mostly specific to FR, and attitudes to them are largely bound up in the ancient empires they controlled in Faerun's antiquity. And so on.

Quote from Darkaiser >>

I'd like to see a game where there was no racial animosity but that means a huge slice out of a big RP pie because there's be less political intrigue, faction fighting, etc.

Factions <> races. It'll take more for the players to learn it, but you can say "The people of the city of Haught feel that the people of the City of Flith are degenerate scum!" just as easily as "The Elves think the Dwarves are degenerate scum!". It just takes more effort on both parts to explain and learn and hold onto the information - it's already a stereotype that dwarves and elves dislike each other, so it's often easier to stick with that than go through all the effort of building a world where that's not the case! But it is doable!

This is a world building question not a question where someone can respond with some essential trait universally held to a species in all settings.  If this is your own setting, you get to pick how race shakes out.  A lot of established settings do reflect stereotypes lifted from fantasy fiction, but a lot of those stereotypes have been interrogated and other established settings don't reflect those stereotypes.  So really unless you're trying to reflect some "canonical" politics and socio-economics in a specific game world, it's up to your imagination.

Humans. No one ever makes content about humans or focuses on them at all. They are so default.

A few more options:

Vampires

Catgirls

The Kardashians

Depends on campaign setting but: Yuan-ti (from a really evil empire), any kind of tiefling (it actually says it in their description), orcs, kobolds, goblins, hobgoblin, drow, duerger, orcs, half orcs, bugbear, the reborn and dhampire have undead traits. Maybe lizardmen since they'll eat humanoids and for that treason thri-kreen if you're an elf. And lycanthropes regardless of race.

I'm thinking of working on a Yuan-ti that is not evil which means I have to write a creative background and probably have his/her snake traits not be obvious...since if you see a yuan-ti pureblood, it'll probably be seen as a spy.

Quote from Foodgeek >>

Depends on campaign setting but: Yuan-ti (from a really evil empire), any kind of tiefling (it actually says it in their description), orcs, kobolds, goblins, hobgoblin, drow, duerger, orcs, half orcs, bugbear, the reborn and dhampire have undead traits. Maybe lizardmen since they'll eat humanoids and for that treason thri-kreen if you're an elf. And lycanthropes regardless of race.

I'm thinking of working on a Yuan-ti that is not evil which means I have to write a creative background and probably have his/her snake traits not be obvious...since if you see a yuan-ti pureblood, it'll probably be seen as a spy.

 If you use the half-breed rules from the Character Origins UA, you could make a half-Yuan-Ti that has all the Yuan-ti traits but much more subtle/human appearance

Monatgues and Capulets.

You don't have to go that far. Yuan-ti's are already written as the most human-like of their kind. Put scales on their arms and hands (or legs or back) and he/she is always wearing armor or gloves/clothes that cover up their snake-like features. Or scales on head (not face) and go the cloak route like Sofina in honor among theives. You can supplement with illusions if you are a caster and has access but you shouldn't have to.