The short story "Liar!" has been referenced as the source for the claim that the robot would be unable to take action, or burn out after doing so. However, this story is mentioned, and in a sense, disproved in a later work - Robots of Dawn.
The relevant quotations are as follows:
As the theory of positronic brains has grown more subtle and as the practice of positronic brain design has grown more intricate, increasingly successful systems have been devised to have all situations that might arise revolve into non-equality, so that some action can always be taken that will be interpreted as obeying the First Law.
Chapter 4 - "Fastolfe"
The robots have improved significantly since the days of Susan Calvin - one should remember that the stories involving her are the stories about the pioneers in robotics, when human-like robots were only being born and perfected.
In Robots of Dawn we're dealing with humaniform robots, so sophisticated they even have all the external human attributes (genitalia included, which is an important detail in that novel). We are also introduced to the concept of "mental freeze-out", which is what happened to Herbie in "Liar!". Doctor Fastolfe, the leading expert on robots at the time the novel is set in, says that such freeze-outs are nigh-impossible with modern day's robots, since they don't only judge matters quantitatively (as the robot from the film did), they are able, in the very, very rare case of exactly equal outcomes, involve randomisation.
“Let’s suppose that the story about Susan Calvin and the mind-reading robot is not merely a totally fictitious legend. Let’s take it seriously. There would still be no parallel between that story and the Jander situation. In the case of Susan Calvin, we would be dealing with an incredibly primitive robot, one that today would not even achieve the status of a toy. It could deal only qualitatively with such matters: A creates misery; not-A creates misery; therefore mental freeze-out.”
Baley said, “And Jander?”
“Any modern robot—any robot of the last century—would weigh such matters quantitatively. Which of the two situations, A or not-A, would create the most misery? The robot would come to a rapid decision and opt for minimum misery. The chance that he would judge the two mutually exclusive alternatives to produce precisely equal quantities of misery is small and, even if that should turn out to be the case, the modern robot is supplied with a randomization factor. If A and not-A are precisely equal misery-producers according to his judgment, he chooses one or the other in a completely unpredictable way and then follows that unquestioningly. He does not go into mental freeze-out.”
Chapter 7 - "Fastolfe"; emphasis mine.
Note that it's not completely impossible to make a modern robot go into mental freeze-out, since that's what the whole plot of the novel is based on. However, it's proven that it requires supreme mastery of the inner workings of the positronic brain and robots' psychology, something only two persons ever are capable of.
The same novel also deals, without naming it, with the Zeroth Law.
R. Giskard kills R. Jander, his robot friend, in order to save humanity (the inhabitants of the Earth) from a sinister spacer plot, and also to save Spacers from interbreeding and dying out in seclusion.
However, the Zeroeth Law is more of an exception than a rule. From what I've read so far, it's only occurred twice. The reason - not every robot is aware of the problems the humanity faces. One could probably speculate that humanity is just a collection of humans to them, while in reality it's a bit more complicated than that. My point is, that is not a situation that would case a robot to invoke Zeroth Law - because it wasn't built into them. The machines in "The Evitable Conflict" were explicitly stated as given access and control of all of humanity's resources and knowledge. They were able to formulate the Zeroeth Law - it doesn't mean every robot can. The characters of Robots of Dawn were also quite exceptional. I wouldn't involve that law in the trolley problem, but if you wish to, what I said above holds.