So, after I watched Dead Again, I had all kinds of great thoughts circulating in my head about this subject... but I hurt my back, and then I got a flu or maybe food poisoning (yes, the drama never stops in the Deerfield house) - and am only just now feeling up to sitting at my desk for more than a few minutes... and I've lost most of my sharp insights.
Nonetheless, I will attempt to touch on the points briefly.
One of the difficulties of the story in Dead Again is that there is a tremendous amount of exposition. Not only do we have a mystery in the present day, but a whole second storyline in the past - plus we have to learn about reincarnation and about hypnosis - at least as they work in the universe of this story.
The primary tool Scott Frank uses to help us over these expository chunks is humor. He has an entire character, played by Robin Williams, whose function is essentially to explain the rules: "Thanks to fate, the only cosmic force with a tragic sense of humor, you burn someone now, they get to burn you later. That's the karmic credit plan. Buy now, pay forever."
Because we're entertained by the speeches, we don't mind them so much.
In fact, there's another comic relief character whose primary function is to bring in exposition: Mike Church's friend Pete. He explains, through a story, how traumatic amnesia works. This scene is at once slightly creepy, amusing, and even a little bit touching. This complexity of tone helps distract us from the fact that we're getting expository chunks.
Another use of humor is as a distraction. When we are introduced to the character who will prove to be the antagonist - he seems like a harmless kook. It is the light humor in the scene which makes this work, because he is, in fact, borderline creepy. He is, in fact, casing the place. He recognizes the period furniture and so likely surmises Mike's connection to his own past. His hypnotism without permission is a violation, but he deflects criticism of his behavior by commenting that the water tastes a bit like bourbon. He picks up the glove (a crucial plot point, the fact that we know he saw it there) but then deflects attention from his action by commenting again on the furniture.
Another writer may have played this introduction of the character much differently, but using humor to deflect in this way works to keep our suspicion off of him for longer.
One more use is the "calm before the storm,"... OK, so this is where my memory fails me. I know that I noted this as I was watching it - but it's been a couple of weeks and I don't recall the moment.
There are even small injections of humor into the middle of the most dramatic scenes: in the final confrontation, Pete walks in with a pizza. Here humor helps to ratchet up the tension one more degree.
While all of these examples are about humor in this drama, the importance is really the modulation of tone overall. The placement of slower scenes before the most intense ones, of a mix of humor and concern and curiosity and romance and danger - all in the same scenes. Rarely is anything played for one note. Rarely is a scene about one element of the story, or one aspect of character development. This modulation helps make such a complex plot work by packing more into each scene.