Generally speaking, commanders did have to take weather into account - carriers couldn't operate aircraft in high seas. Ships could also take damage in heavy storms. As a result, both sides sought to avoid battle in stormy weather, and it was taken into account for tactical planning. However, typhoons tend to form and move too quickly for them to interfere with strategic planning.

Minor weather events affected several battles. During the Battle of the Coral Sea, the Japanese carrier Zuikaku was hidden by a rain storm at the time Allied carrier aircraft arrived to attack the Japanese fleet. This resulted in the American aircraft targeting it's sister ship, the Shokaku. No damage was taken by the Zuikaku as a result. The Americans had a similar experience at the Battle of Santa Cruz - during the first Japanese strike, the USS Enterprise was hidden by a squall, causing the Japanese to focus on the USS Hornet. The Enterprise did, however, take heavy damage in the second Japanese strike. At Midway, high level cloud may have resulted in the failure of the Japanese scout aircraft Chikuma No. 1 to detect the US Task Force 17 early enough for the Japanese to respond.

One campaign was seriously affected by poor weather. The Aleutians Island campaign saw both sides being seriously hampered by severe polar weather. The opening action, the Japanese raid on Dutch Harbor on June 3rd 1942, saw many of the attacking Japanese aircraft losing their way, and being forced to return to their carriers or ditch at sea. American counterattacks were similarly foiled. The American assault on the island of Attu ran into logistical difficulty due to poor weather. Troops on Attu also suffered due to extreme cold, with 1200 men being evacuated with cold-related injuries, 31% of the total casualties. The Allied invasion of Kiska ran into similar problems, but this became farcical once it was discovered that the Japanese had abandoned the island during a period of heavy fog.

In December 1944, Admiral Halsey accidentally led Task Force 38 into a typhoon off Luzon. While Halsey should have recognised the signs of the impending typhoon, he had been given poor weather forecasts. These suggested that the typhoon, dubbed Typhoon Cobra, would miss his force. The fleet had been undergoing refuelling operations, which meant that many of the ships, especially the escorting destroyers, were top-heavy. Three destroyers would capsize and sink, while a heavy aviation fuel fire was started by runaway aircraft aboard the light carrier USS Monterrey. Damage was taken by most of the ships in the fleet. Halsey would again lead a fleet into a typhoon, Typhoon Connie, in June 1945. As Connie was weaker, no ships were lost, and few casualties were taken, although many ships incurred damage.

Sources

Parshall, J., Tully, A., Shattered Sword, Potomac Books, 2007

Toland, J. , Rising Sun: The Decline and Fall of the Japanese Empire 1936-1945, Pen and Sword Military, 2005

Hornfischer, J. D., Neptune's Inferno, Presidio Press, 2012

Burks, R. E., Logistics Problems on Attu, ALMC, found online at http://www.almc.army.mil/alog/issues/MayJun03/MS778.htm

Not sure if this counts as a "battle" but weather played a major role in bombing strategy against the Japanese homelands. It was especially important in the atomic bombing operations, because they were required to be visually targeted (as opposed to using radar). This meant the Hiroshima raid was pushed back to August 6th (as opposed to August 3rd) and the second bombing raid was pushed up to August 9th (and even then, Kokura was the first target for that raid, but cloud cover and maybe smoke obscured the target, making Nagasaki the fall-back target). So weather is one of the reasons that the bombings were so back-to-back; it pushed the first bombing later and the second bombing earlier, so little time in between that the Japanese were barely able to confirm that Hiroshima had been an atomic bomb before the second bomb came.