Serious and Comic (Largely Depending on the Character), But Mostly Objective
Compare Speedy ("Runaround") and Herbie ("Liar!"): both robots are caught in dilemmas where the Three Laws are in conflict. Speedy has to follow an order but also has to protect himself; Herbie has to not hurt someone, but doing so might hurt someone else. So, they're in similar situations.
Now compare how they're dealt with: Speedy is compared to a drunk and spends his time singing show tunes; Herbie sits alone in his room, reading books about heartache and romance. Both those situations are kind of silly, but doesn't Herbie's situation seem a little more serious and tragic? And doesn't Speedy's situation seem a little funny—a drunken robot singing show tunes? (Asimov could have shown a robot malfunctioning in lots of ways, but he chose to have the robot sing. Everyone looks a little silly when singing, especially show tunes.)
This is why we say that Asimov's tone can be serious or comic largely depending on the character. For instance, Powell and Donovan are often a little ridiculous; but Calvin is never treated as the object of humor.
Still, the overall tone of Asimov's work tends to be pretty objective. Powell and Donovan may be comic figures at times, but that's largely because they seem a little silly (what with all their fighting and joking around with each other). We never get the impression that the story is really on one side or the other. The story may make us root for Calvin to figure out the problem, as in "Liar!"; but when she drives Herbie insane, the narrator doesn't tell us that it's OK and she had her reasons to. Rather, the narrator doesn't tell us what to think and lets us come to our own conclusions.