How much responsibility does ATC have to alert a flight about being in a dangerous situation?
This is potentially a very broad topic, but here's some information about the US.

As a general principle, safety is ATC's first priority so obviously they have a basic responsibility to provide whatever information and action are required to keep aircraft safe:

FAA ATC orders 2-1-1:


  The primary purpose of the ATC system is to prevent a collision
  between aircraft operating in the system and to provide a safe,
  orderly and expeditious flow of traffic, and to provide support for
  National Security and Homeland Defense.


The orders explicitly say that ensuring separation and issuing alerts take priority over everything else. Section 2-1-2, DUTY PRIORITY:


  Give first priority to separating aircraft and issuing safety alerts
  as required in this order. Good judgment must be used in prioritizing
  all other provisions of this order based on the requirements of the
  situation at hand.


Section 2-1-6, SAFETY ALERT defines what an alert is and when to issue one (this section is referenced throughout the orders):


  Issue a safety alert to an aircraft if you are aware the aircraft is
  in a position/altitude that, in your judgment, places it in unsafe
  proximity to terrain, obstructions, or other aircraft. Once the pilot
  informs you action is being taken to resolve the situation, you may
  discontinue the issuance of further alerts. Do not assume that because
  someone else has responsibility for the aircraft that the unsafe
  situation has been observed and the safety alert issued; inform the
  appropriate controller


It's difficult to say much more than that because there are so many possible scenarios. But you can search for "safety alert" in the orders and the AIM to get more information and there's some limited content in the orders 5-15 about radar displays and alerts.

If ATC fails to issue a safety alert and that leads to an accident, then it will be mentioned in the NTSB report, e.g. LAX04FA205:


  Contributing to the accident was the failure of both the Center and
  TRACON controllers to properly respond to the aural and visual Minimum
  Safe Altitude Warning System (MSAW) alert.


But actually acting on a safety alert is entirely the pilot's responsibility, of course (this is from the AIM 4-1-16):


  Once the alert is issued, it is solely the pilot’s prerogative to
  determine what course of action, if any, to take