This has also been covered somewhat in the animated movie Justice League: Doom, where Batman's computer is hacked by Mirror Man after hiding in the rear view mirror of his Batmobile. Cyborg plays a role in this movie, but he is not a member of the Justice League during the movie, and so is not included in the plans to take advantage of each member's weakness. Aquaman does not appear in the film at all.
Mirror Man uncovers Batman's files on the Justice League's weaknesses, including plans to neutralize them, while acting on orders from Vandal Savage. Vandal Savage modifies these plans to permanently eliminate the Justice League, and then enlists the help of several others to carry out the plans: Cheetah, Star Sapphire, Mirror Man, Bane, Ma'alefa'ak, and Metallo.
Superman is lured to a man holding a gun to his own head on a rooftop, surrounded by a crowd below. The man turns out to be Metallo in disguise. Little did Superman know, the gun Metallo was holding contained a kryptonite round. Metallo shoots Superman with the gun and he falls from the roof to the ground, unable to be operated on by first responders due to his skin being impervious to their medical tools. From this we can deduce that Superman's weakness is kryptonite. We know from other material that Superman is also weak to magic, though that weakness is not exploited in this movie.
The Flash is lured to a train heist, where Mirror Man has devised a special cage with a woman trapped inside. Flash can reach in and open the cage, but in so doing, triggers a metal spike that gets shot through Flash's wristbone. Attached to this spike is a bomb with a timer and an accelerometer; if Flash does nothing, the bomb will explode. If Flash runs off, and then slows down at all when he's at a safe distance, the bomb will explode. From this we can deduce that The Flash's weakness is his urge to help others, even at the cost of his own safety/life.
Wonder Woman is lured to a crime scene on a dock where Cheetah lies in wait. Cheetah has nanobots of some kind painted onto her claws in a form of high-tech poison. When this nano poison enters Wonder Woman's bloodstream via a scratch attack from Cheetah, it causes her to hallucinate and see everyone around her as Cheetah. She therefore goes on the attack against everyone she sees. This is dangerous to Wonder Woman because, due to her warrior upbringing, she won't stop fighting until she killed or her heart gives out due to exhaustion/exertion (or wins). From this we can deduce that Wonder Woman's weakness is her lust for battle and disdain for giving up a fight, which could lead to her death by a superior foe or exhaustion.
The Martian Manhunter is in his human form and out to drink at a bar on the pier with coworkers. A lovely woman (Ma'alefa'ak shape-shifted in disguise) offers him a drink, which he accepts, only to find out that it is laced with magnesium carbonate (a chalky salt). This compound is harmful to Martian biology, and highly flammable, even in water. Ma'alefa'ak uses a lighter to set the Martian Manhunter on fire. This combination will keep him ablaze for days, if not weeks, with no apparent way to put out the fire, and no one around knows anything about Martian physiology to offer any help. From this we can deduce that the Martian Manhunter's weakness is fire. Especially chemical fire.
Green Lantern is lured to a hostage situation at a mine, where he first attempts to negotiate the miners' release, and then attempts to rescue them by force. In the process, one of the hostage takers is able to detonate a bomb, killing all the miners, including a woman that looks a lot like Carol Ferris (who becomes Star Sapphire, and who was the love interest of Hal Jordan, this movie's Green Lantern). This causes Green Lantern to question his ability to do his job to protect the galaxy and those in need, and to doubt his own willpower (the trait that gives Green Lanterns their strength). Little did he know, the mine was filled with thin vapors of a refined form of Scarecrow's fear gas, and the miners were all robots designed to look like humans. From this we can deduce that Green Lantern's weakness is fear, or anything that can shake his will to continue fighting.
Batman did not have a plan to neutralize himself stored on his computer in the Batcave, so the plan enacted to kill him must have been one of Bane's and/or Vandal Savage's making. A gardener rushes to the Wayne Manor to tell Alfred and Bruce Wayne (Batman) that someone has vandalized/destroyed the mausoleum holding Batman's parents. Batman is immediately moved into a frenzy and rushes out to the mausoleum. The gardener then reveals himself to be Bane, and proceeds to beat the crap out of Batman, who is both completely unarmed/unprepared and also still in psychological shock after seeing his parents' resting place so desecrated. Bane knocks Batman unconscious, and then buries him alive in his father's casket. From this we can deduce that Batman's weakness is his psychological trauma from the past, specifically involving his parents and their death. Of course, we all know the real truth is that Batman has no weakness, so he was able to overcome Vandal Savage's plans and save the team from their own potential demises.
Since Cyborg and Aquaman are not covered by the movie, I can only posit that Cyborg, being a cyborg and largely an electronic being, could be disabled via an EMP or some kind of advanced computer virus/nanobot technology. Aquaman probably overestimates and isolates himself, seeing himself as above others due to his status as King of Atlantis, whose domain covers nearly 75% of the Earth's surface, and covers much more depth than the realms of Earthlings.
All the Justice League mentioned above are susceptible to magic, especially Superman. Without Doctor Fate or Zatara (or Zatara's daughter) on the team, they don't have much of a defense against magic. Shazam (Captain Marvel) utilizes magic, but as far as I know it's only to transform from his normal child self into Shazam, and to call down magic lightning bolts.