There is no love between the NBA's Paul Pierce (Boston Celtics) and LeBron James (Cleveland Cavaliers/Miami Heat), particularly due to the frequent on-court duels between the two. One even went to the full seven game series, with Pierce ultimately winning and getting a ring before James. LeBron's losses to the Celtics was one of the biggest reasons as to why he left to form a Big 3 with Dwyane Wade and Chris Bosh in Miami, to counter that of the Celtics' Big 3 of Pierce, Kevin Garnett and Ray Allen. It got so bad that James exclusively referred the Celtics as "that team", and Pierce certainly doesn't bother with showing any sign of sportsmanship with James (e.g. no handshakes, no hugging, barely talks to James off the court and so forth).
It goes back even further than 2008. They got in a few verbal confrontations during LeBron's rookie season, and once nearly got into a fistfight in the locker room over a pre-season game prior to LeBron'ssophomore year.
There is arguably even less love between Kobe Bryant and Shaquille O'Neal—The Other Wiki has its own article on their relationship, or lack thereof.
Things have improved between them in recent years.
Don Cherry and European players, though he's softened in recent years.
Many British football clubs have these.
Manchester United vs. Manchester City
Liverpool vs. Everton
Celtic vs. Rangers
Arsenal vs. Tottenham
Aston Villa vs.Birmingham City
Southampton vs. Portsmouth
Watford FC vs. Luton Town- proving that it's not just the more well-known clubs who hate each other's guts.
This trope is probably played straightest in boxing, not surprisingly considering the 1-on-1 aspect of the sport. The most famous rivalry was between Muhammad Ali and Joe Frazier, with Frazier still bitter long after their respective retirements. This was the classic form of rivalry, as they were completely opposed in the ring and out in terms of temperaments, attitudes, and persona.
Some sports rivalries, including 99% of all college sports, don't include the "respect" part. In the case of rivalries like Ohio State-Michigan, Duke-North Carolina and this troper's Arizona-Arizona State, there's nothing but pure hatred. "Sports rivalry" just has a better ring to it than "sports arch-enemyship".
No UCLA and USC? Sometimes they make fun with their opponent's mascot.
Alexei Yagudin and Evgeni Plushenko, possibly the fiercest rivalry in the history of figure skating.
Soccer-wise, there's a well established rivalry between Brazil and Argentina.
There are also others not quite as famous as Brazil-Argentina (although it depends on where you're from), but notable others include Portugal-Spain, Holland-Germany, England-Argentina and Mexico–USA.
Regarding Brazil and soccer, the local teams have plenty. Most cities have 2 major teams that hate each other. Recife has 3. Rio and São Paulonote though one is not in the city itself, but nearby Santos each has 4!
The NFL is littered with rivalries. Some date back to the league's early days while others are more recent, but they all share the same aspect of adding an extra layer of nastiness to an already physical and violent sport. Perhaps the most intense and bitter rivalries are between divisional opponents, such as: Green Bay Packers-Chicago Bears, Dallas Cowboys-Washington Redskins, and Pittsburgh Steelers-Baltimore Ravens.
Major League Baseball:
New York Yankees vs. Boston Red Sox, the most famous (and possibly most terrifying) of them all. In terms of World Series titles, it's incredibly lopsided. The Red Sox were one of the top teams in the early days of baseball, having won 5 of the first 16 World Series played (to the Yankees' 0)...and then the Yankees managed to win 26 before the Red Sox got their 6th, which came in the 100th World Series played. (It now stands at 27-8). Even with the lengthy drought, however, the Red Sox' 8 titles stands as the fourth-most. (The two are also both among the most successful at winning the World Series when they get there—the Yankees winning 67.5% of World Series they've been in, the Red Sox 66.67%. Only the Pittsburgh Pirates, who are 5-for-7, and a few expansion teams that have 100% success rates but only 1 or 2 appearances, are better.)
Los Angeles Dodgers vs. San Francisco Giants, the West Coast variant and arguably even more passionate and glorious.
St. Louis Cardinals vs. Chicago Cubs, the Midwest version, which splits the state of Illinois, fanbase-wise. Another lopsided rivalry, where the Cardinals have 11 World Series titles (the most recent from 2011) and the Cubs have two (the most recent from 1908). This one is a bit less vicious than the Yankees/Red Sox rivalry, even bordering Friendly Rival territory. It's mostly teasing, especially from Cardinal Nation's stand point. As former Cardinals pitcher-turned-broadcaster Al Hrabosky once said, "Any team can have a bad century."
Larry Bird and Earvin "Magic" Johnson faced each other in the 1979 NCAA Men's Division I Basketball Championship game (still the highest-rated game in the history of televised college basketball). They both entered the NBA that year, each landing with one of the NBA's bell cow franchises (Bird with the Boston Celtics, Johnson with the Los Angeles Lakers). Between 1980 and 1991, eleven of the twelve championship finals featured Bird's Celtics' (three wins) or Magic's Lakers (five wins), meeting each other three times (the Lakers's taking two of three). The Bird/Magic rivalry is often credited from saving the NBA (before Bird and Magic, the NBA finals were notorious for being shown on late-night tape delay).
Lakers-Celtics is a similar rivalry, as both are the most successful teams of the league - 17 championships for the Celtics, 16 for the Lakers, are located in opposite coasts (though it started locally, as the Lakers were in Minneapolis) and made a staggering 12 finals against each other (9-3 to the Celtics so far).
Tom Brady versus Peyton Manning. You could really go either way on this one. As of October 5th 2013, their regular season passer ratings are virtually identical, as well as their significantly lower playoff passer ratings. Brady holds the edge in championships and win percentage, while Peyton has more MV Ps and All-Pro nods. The consensus among people that examine their entire careers is that Brady usually has a better defense, but Peyton has a better offensive supporting cast (though on occasion neither has been true). The rivalry mostly exists between the two teams that have the other player; Brady and Manning themselves have a great deal of respect for each other.