Has there ever been a father and son in the NFL/NBA/NHL/MLB who were both active players at the same time?
Absolutely. Ken Griffey Jr. and Ken Griffey Sr. played together in 1990 for the Seattle Mariners. They once hit back to back home runs in a game.
They played about a month together (8/31 to the end of the season).
In Hockey, Gordie Howe played with his sons Mark and Marty in 1979-80 for the Hartford Whalers. Wikipedia has a nice page with a list of familial relations in the NHL (they seem to have more than most leagues, probably because the sport is less popular so more likely for familial relations to be successful). The direct link to the Howes' is at Legends of Hockey.
In the NFL, it doesn't look like anyone really did in the modern era due to the very short career length compared to other sports (except for kickers); in the 1920s, Ted and Charles Nesser played together, Ted as player-coach, and I don't doubt a few more similar occurrences would be found if you looked through the 10s and 20s.
George Wilson coached his son (also George) for the 1966 Miami Dolphins, the only example of that I found. Ed and Brad Budde were the closest to playing with each other from the list I saw, Ed playing 1963-1976 for the Chiefs and Brad joining the Chiefs in 1980-1986. This is from a list of father-son relationships from the Pro Football HOF.
From the comments, Leatherwing noted that Wikipedia also has a good NBA Father/Son page; Kobe Bryant nearly overlapped with Bean, his father (1992/1996).
In Cricket there are several examples of that: 18 times a father and son have played in the same match
In 1851 and 1853 even William Lillywhite has played with his two sons: John and James
I don't know if car racing meets the definition of a sport in the OP's question, but there have been some contemporaneous fathers and sons in car racing. Perhaps the most famous is Dale Earnhardt Sr. and Jr. They were in the same race when Earnhardt Sr. died.
In (international) football: Eiður and Arnór Gudjohnsen.
On 24 April 1996, he and his father entered football history in an Iceland international friendly in Tallinn. Arnór started the match, and Eiður came on in the second half as a substitute for his father.
This weekend 55 year old Dave Beasant was included on the bench for Stevenage FC as a second choice goalkeeper, in place of his injured son Sam who is also a Stevenage goalkeeper. So they will never appear in the same team, or even in the same squad, but are both currently 'active' players for their club.
Also another recent case in football was Brazilian legend Rivaldo playing for the same team as his son, Rivaldinho. They both played for Brazilian team Mogi Mirim earlier this year.
In ice hockey, Sami and Kasperi Kapanen played last season in the Finnish league.
It's actually fairly common in U.S. auto racing; teams in those sports are often family owned and passed down between generations.
- Lee Petty raced against two of his sons, Richard and Maurice, in the same NASCAR race in 1960.
- Mario Andretti raced alongside his son Michael in the CART Indy Car series for 8 years, from Michael's rookie year in 1984 (Mario having rejoined the Indy Car series from Formula-1 in '82), to 1992 when Mario retired for good (becoming Michael's crew chief). From 1990 to 1992, Mario and Michael were joined on the circuit by Jeff, Mario's son and Michael's brother, who raced in CART until a serious crash in 1994 effectively ended his career. Michael, in turn, practiced alongside his own son Marco before the latter's first Indy 500 in 2006, but the two never competed in a race as Michael had officially retired in 2002.
- Al Unser Sr and Jr were both in the American Championship (aka CART) for 11 years before "Big Al"'s retirement in 1994. Al Sr and his brother Bobby Unser also raced together.
- Dale Earnhardts Jr and Sr competed together in the Winston Cup Series in 2000 and 2001 before Sr's fatal crash at Daytona.
In motorsports, Jan Magnussen is still an active competitive driver in Danish touring cars while his son Kevin drives for McLaren in Formula 1. This may not count since they are very different series.
Martin Brundle and his son Alex Brundle competed in the same team, Greaves Motorsport, driving at the 24 hours of Le Mans in 2012: http://en.wikipedia.org/wiki/Alex_Brundle
According to this FIFA article also Carlos Alonso and Juan Carlos Balazar play in the same official match in first division Peruvian football league
Also in curling is possible to find some matches where father and son play in the same match:
- On this article from Milwakee Journal (date 27 feb 1937!) there is the news where in two matches there where fathers and sons: George La Marre played versus his sons William, Edwards and Charles and on another rink Lachman MacDonald played with his son Bruce
- On november 2011, according to this article the famous Glenn Howard played versus his son Scott
Conrad Harrison and his son Dean Harrison both won TT titles in the same week.
According to Wikipedia:
With his father Conrad Harrison winning the Sidecar Race 1 and Dean Harrison's Lightweight win this was the first occasion of father/son winners in different classes during the same race week.