O'Brien himself has not died during the course of the Star Trek TV franchise. However, he has experienced some near-misses in DS9.
During the DS9 episode "Whispers", a clone of O'Brien was created by a group of alien rebels in order to infiltrate Deep Space 9 and stop a peace delegation that was to be held there. The clone turned out to be too perfect and believed that it was the real Miles O'Brien, and became suspicious of his colleagues, friends and even his wife when they all began to act strangely to him. This was because the DS9 crew were already aware that O'Brien had been replaced and were monitoring his actions in order to ensure the Perandan peace talks went smoothly and because they would need to keep the clone alive in order to find the real O'Brien.
The O'Brien replicant later escaped DS9 when the crew attempted to take him into custody. He took a runabout to the Perandan system where the whole series of events began and he discovered that his crewmates were waiting for him in a cavern on Perandan II. The clone shot at a Perandan guard but another guard retaliated in kind and the clone was mortally wounded. A door opened in the cavern, revealing the real O'Brien attached to some kind of medical equipment who was being seen to by Dr. Bashir. The real O'Brien got up and watched as his clone died in front of him.
In the third season episode "Visionary", Miles experiences periods of temporal displacement. Multiple times throughout the episode, he shifts forward a few hours into the future, and in one such time shift he witnesses his own death. His future self opens a panel in a corridor and is suddenly shot dead by a laser device planted inside the panel. After the experience, O'Brien informs the crew of the occurence and security is tightened around the area where O'Brien saw himself die.
Even though O'Brien avoids being murdered, another time-shift reveals that he still dies when the radiation he was exposed to at the start of the episode causes lethal damage to his brain. Luckily, before he shifts back into the past, Dr. Bashir tells O'Brien to inform the past Bashir of the right kind of scan to find the abnormalities in Miles' brain and treat them before it's too late.
Before O'Brien's treatment fully kicks in, he experiences one more time shift where he witnesses the destruction of Deep Space 9, but not how it was caused. After the shift, he comes up with the idea of triggering another time-shift to a point before the station explodes. He uses a device that injects his body with the same radioisotopes that have been causing the shifts, knowing that the toxic isotopes will kill him if he doesn't hurry.
The cause of O'Brien's time-shifting is revealed to be an artificial quantum singularity from the engine core of a cloaked Romulan warbird orbitting DS9. The tetryons given off by the Romulan ship's warp core react with the radioisotopes affecting O'Brien, causing the time-shifts. During his last time-shift, Miles alerts his future self and the two head into Ops when suddenly the Romulan warbird decloaks and opens fire on the station. Now that Miles knows what's happening, he can go back and prevent the station's destruction, but the injector device O'Brien is wearing has poisoned him and he is near death. He gives the device to his future self who takes his place back in the past while the past O'Brien dies from radiation poisoning.