Depends on what nation you're referring to. You also have to remember that professional sports were far less prevalent then than there were today. No Premier League or Champions League. No NBA.
Japan did draft its professional baseball players (baseball was an exceptionally popular sport in Japan, even during WWII). By 1945, virtually every professional ballplayer in Japan had volunteered or been conscripted. Almost 70 of them died during the war.
In the United States, professional athletes were drafted. Some like Red Sox slugger Ted Williams volunteered. Now, only a minority of these athletes ever served in combat (although two Major League players did die in combat during the war). Many played exhibition matches in US Army- and Navy-run baseball teams as morale-boosters for American soldiers.
Sources: *Japan At War, Time-Life Books (1981)
So many football players either joined the military or were drafted, that in 1943 two teams, the Brooklyn Dodgers football team and Cleveland Rams had to suspend operations for one year and the Pittsburgh Steelers had to merge with the Philadelphia Eagles for one season. In 1944, the Steelers had to combine with the Chicago Cardinals, because neither team could fill their thirty-five man rosters.
Sammy Baugh and Sid Luckman were the two star quarterbacks of the early 1940s. Sammy Baugh was not drafted, nor did he enlist. Because there was such a shortage of young athletic men in the United States at the time, Sammy Baugh had to play quarterback, punter and corner back. Sid Luckman enquired about joining the US Navy, but since he was a law school graduate that had passed the bar exam, the Navy wanted him to become a JAG. That did not interest Luckman, so he declined to enlist. He was not drafted either, but in December, 1943 Sid Luckman joined the US Merchant Marine.
In baseball, the two biggest stars of the early 1940s were Joe DiMaggio and Ted Williams. DiMaggio enlisted in the US Army Air Force 2/17/1943 and spent WWII playing center field for a 7th Air Force baseball team in Oahu that routinely creamed various Navy baseball teams. Ted Williams joined the US Navy on 5/22/1942 and became a pilot with the Marine Corps. Despite doing very well in training, he was not deployed over seas until May 1945 and flew no combat missions during WWII. However, Williams did join the US Navy Reserve and was called up at the age of 33 and sent to the Korean war in 1952. He flew 39 combat missions in Korea.
Gordie Howe was born in 1928 and was to young to be drafted. However, so many NHL hockey players enlisted, it opened the way for him to play for a minor league farm team of the Detroit Red Wings, at the age of 16. The greatest hockey player of the 1940-1960 era was Maurice Richard. He broke his ankle in 1941 and was deemed unfit for military service. He eventually joined the Canadian Merchant Marine in late 1944, despite having many obstacles placed in his path. He had to learn how to become a machinist mate, before the Canadian Merchant Marine would let him ship out.
Finally, there is the example of Joe Louis. the Heavy Weight Boxing Champion. He was drafted by the US Army, but was used to appear at War Bond Drives and show up at state side Army bases to build morale. He was not deployed in any combat areas.
Shout out to Louie Zamperini who was an American Olympic runner in the 1936 Berlin Olympics, and went on to be an air force captain who survived months adrift in the Pacific and then a POW, as chronicled in the book Unbroken: A World War II Story of Survival, Resilience, and Redemption, which I really enjoyed.
I hope this meets the standards of the sub. I'm excited to be able to contribute something!
A similar question was asked a while back, and after spending far to long looking for it, I have the link here. I posted an answer regarding how the Canadian government dealt with the National Hockey League and the players, which I will repost here:
Regarding the NHL, they had an extra two years to figure out what to do, being Canadian-based (even with 4 of 6 teams operating in the US). There is an excellent article by J. Andrew Ross, "Arenas of Debate: The Continuance of Professional Hockey in the Second World War" included in the book Coast to Coast: Hockey in Canada to the Second World War (edited by John Chi-Kit Wong). But I'll try and keep it focused on what happened in the US.
After the US entered the war, there was talk to suspend play. However, as they had argued with the Canadian government previously, the NHL claimed they helped boost morale and benefited the war effort. Frank Calder, president of the NHL until his death in 1943, and then his successor Red Dutton, argued that the total number of hockey players, in all leagues, was less than 400, and mainly Canadian; therefore they had no real effect on the war effort. In addition the teams were indirectly benefiting the governments (a major point Calder used was that by having Canadian players work in the US and be paid in US dollars, they were greatly benefiting the foreign exchange for Canada by bringing in badly needed US currency). Because of these reasons, the league was allowed to continue, but its worth noting that they abandoned the 5 minute overtime period in November 1942 to help alleviate travel for both the teams and fans.
Like the other sports, many hockey players also signed up for the war. Perhaps most famously, the top three players on the Boston Bruins, the so-called "Kraut Line" of Milt Schmidt, Bobby Bauer and Woody Dumart (they were all from Kitchener, Ontario, which was known as Berlin until 1916 due to the large German population) all enlisted in the RCAF; they actually kept playing "amateur" hockey for the air force, helping them win the Allan Cup (the amateur championship of Canada) in 1942. Other players from teams also joined, including Bruins goalie Frank Brimsek, who joined the US Coast Guard. The New York Rangers had the most players enlist, so many that in the 1943-44 season their coach, 42 year-old Frank Boucher, played in 15 of the teams 50 games (he had previously played for the Rangers, but retired in 1938).
Its worth noting that there was no consensus amongst the NHL teams on how they went about dealing with the war. For example the Montreal Canadiens found jobs for many of their players in local factories, therefore keeping them exempt from the draft. This angered other teams, particularly the Rangers and Bruins, who felt they were doing more for the war than the Canadiens and the other Canadian team, the Toronto Maple Leafs (even though the Leafs owner, Conn Smythe, was an ardent military supporter who had served on the front in the First World War and enlisted again in the Second, age 44 in 1939. He also encouraged his players to do the same, and was actually wounded in France in 1944).
Overall though, the NHL played every season during the war, even with depleted rosters. They did lose one team, the New York Americans in 1942 (who called themselves the Brooklyn Americans in their final season 1941-42, even while still playing at Madison Square Gardens), but this was more related to the Depression than the war.
Interesting question. Why might they be exempt?
Enough to provide opportunities for people who otherwise wouldn't have been able to play. One example is Pete Gray, who only had one arm:
"Playing in 77 games for the 1945 Browns -- at a time when major league rosters were depleted because of players' wartime military service..."
http://www.nytimes.com/2002/07/02/sports/pete-gray-major-leaguer-with-one-arm-dies-at-87.html
After the war many of the regular players returned to sports:
"Gray’s major league career ended that same year when many of baseball’s stars returned from the battlefront and assumed their previous positions on the diamond."