Is it ever mandatory to call the police or other emergency service for people? For example, say a friend has suicidal thoughts. Could you ever be breaking the law by not calling the police in such a situation?
A person may be a "mandatory reporter", which means that in some jurisdiction, if you have a certain relationship to a person suspected to be "problematic", in a particular manner, you may be required by law to report those facts to a particular authority. There are a lot of "ifs" in these requirements. For instance, in Washington, RCW 74.34.020 defines a set of "mandated reporters", for abuse of vulnerable adults. The trigger conditions are "abandonment, abuse, financial exploitation, or neglect", and a mandatory reporter reports to the Department of Social and Health Services. I'm not a mandatory reporter, but school personnel and health care workers (inter alios) are. This document summarizes the various legal requirements by state.
In Oregon, a hospital is required to report an attempted suicide, and you report to the Oregon Public Health Division. An exhaustive search of all such laws would be impractical, but based on the laws that I reviewed, there are some general features of mandatory reports. First, the person required to report has to have a special duty of care, for example they are law enforcement, a health care professional, etc. and these laws do not impose a requirement on everybody. Second, the individuals that they are supposed to report about are "vulnerable", however that is defined by law – usually, the implication is that they are not in a mental state that they can take full care of themselves. Third, the report is made to some supervisory social services agency, which is then responsible for taking further action (to prevent the harm). However, health care workers in California are required to report suspected domestic violence to the police. It is unlikely that any jurisdiction in the US requires reporting suicide to the police, since suicide is not a crime in the US, and the police are not a social service agency, they are a law enforcement agency (no law is being broken).
In the US there are very few situations where any requirement to help or report a situation exists. Some people, for example certain professionals taking care of children, have the status of “mandatory reporters” of child abuse allegation or evidence.
You might remember in the final episodes of Seinfeld the group is arrested for mocking someone in trouble rather than helping. They apparently ran afoul of a local, odd ball duty-to-rescue law. This is an anomaly.
In the Wikipedia page for duty to rescue you will see that in common law countries there is no such duty in general. Exceptions are in cases where you caused the hazard in the first place or for certain special relations between you and the person in trouble.
Requirements to report crimes are relatively rare. One common example involves money laundering.
For example, in england-and-wales, Section 330 of the Proceeds of Crime Act 2002 requires, under specific circumstances, a person in the "regulated sector" to report suspicions of money laundering to a nominated officer or a person authorised by the National Crime Agency. In the case of the former the nominated officer in turn is required to report to a person authorised by the NCA, pursuant to Section 331. Schedule 9 defines what is meant by the regulated sector.
While "breaking the law" typically suggests something like a criminal offense, there are other considerations not mentioned above where failing to call for help could give rise to legal consequences.
Anytime someone else could have legal liability to you for compensatory damages on almost any theory, you have an obligation to mitigate damages, and so, if you could potentially mitigate the harm to yourself and you fail to do so, you could be denied recovery for avoidable damages.
Also, while only a few states have a "duty to rescue", in situations where you put someone in harms way through your actions, or deny them an ability to take care of themselves (e.g. by putting them in custody) you have a duty to make things right by any means possible. Similarly, if someone is entrusted to your care (e.g. you are a babysitter) you may have an affirmative duty to seek help.
In this case, I wouldn't call the police, but a help organization instead. Your friend needs to talk to a psychologist, not the police. In most countries, there are organizations that offer support by phone for such situations. The Wikipedia article Crisis Hotline may be of some help. It even contains phone numbers for the US.
As far as I know, most countries require you to help or call for help if you see an emergency (or suspect that some bad thing is about to happen). They don't require you to call the police.