Why did The Dude hate The Eagles so much?

Out of universe, this is probably T Bone Burnett talking. From IMDb:

T Bone Burnett acted as music consultant for the movie, and helped Joel Coen and Ethan Coen establish the Dude's taste in music. Burnett selected many of the existing songs in the movie, and also suggested the Dude's hatred towards The Eagles (Burnett himself is not a fan either). One of the band's member, Glenn Frey, was reportedly so dismayed about this that he once even angrily confronted Jeff Bridges when they met at a party.

Says Burnett:

"[The Eagles] sort of single-handedly destroyed that whole scene that was brewing back then."

On a side note, the Dude's aversion to the Eagles also helped the team to secure a desired song, saving them as much as $150,000. From Wiki:

[Burnett] had a tough time securing the rights to Townes Van Zandt's cover of the Rolling Stones' "Dead Flowers", which plays over the film's closing credits. Former Stones manager Allen Klein owned the rights to the song and wanted $150,000 for it. Burnett convinced Klein to watch an early cut of the film and remembers, "It got to the part where the Dude says, 'I hate the f@$&in' Eagles, man!' Klein stands up and says, 'That's it, you can have the song!' That was beautiful."

It's possible a hardcore, counter-culture 60s hippie like the Dude would consider the Eagles sellouts. From The Big Lebowski: An Illustrated, Annotated History of the Greatest Cult Film of All Time:

Even the choice of Creedence represents the more rebellious side of classic rock. [...] Creedence symbolizes the polar opposite of the band the Dude openly hates, the Eagles. In the mid-1990s, the Eagles became the first mainstream rock band to sell tickets for more than $100. If Creedence had a dash of antiwar insurgency, the Eagles were all Baby Boomer contentment, a feeling encapsulated in the song "Peaceful Easy Feeling". [...]

Eagles fans should forgive the Dude for reacting so harshly to [that song]. It was, after all, the Dude's perceived lackadaisical hippie way of life that Jeffrey Lebowski attempted to exploit, and the lazy-day vibe of the Eagles' "Peaceful Easy Feeling" [...] is hippiedom as filtered by Boomer nostalgia. By this time in the film, it's been made clear that there's far more to the Dude than any peace-and-love cliche.

I think it just makes the scene work. Consider: An angry and aggressive cab driver is listening to "Peaceful, Easy Feeling" (this is funny all by itself), The Dude had a hard night and "Peaceful, Easy Feeling" is making it worse (??). After requesting that the cab driver turn it off, the driver throws him out so he can keep listening to, again, Peaceful, Easy Feeling.

It's very much in line with the comedic style of the whole film: subtle absurdity that can escape you the first few times you see it.

This answer requires zero outside knowledge of The Eagles, a fact which I think makes it more compelling, not less.

I'm from New Jersey. Specifically I grew up two towns away from Asbury Park, the town (which I live in and am typing this from) which is synonymous with Bruce Springsteen. He is a legend here. Like the Pope of New Jersey. I grew up thinking he was overrated, probably because my parents and people their age loved him, so I was being 'A Rebelious Teen'.

My guess is that The Eagles held a similar stature of being The California Band like Bruce is the New Jersey Guy. To not be a fan of The Eagles would probably be akin to being against California, such as the vile I felt from my fellow New Jerseyans for not being a fan of The Boss while growing up. This would explain the taxi driver's reaction to The Dude saying he hates The Eagles. I would absolutely get thrown out of most cabs, bars, etc, if I was vocal about my disdain of Bruce Springsteen.

As I'm probably The Dude's age, like, right now, I can tell you I've come to appreciate Bruce and his songs, mostly because of what he's done for the area, but "I'm On Fire" is definitely favorited on my Spotify. Maybe The Dude would have eventually softened on The Eagles, at least that's what I hope.