Go through these in order and stop at the first one that applies. If you are reading this for someone else, answer for them.
Call emergency services after a major injury if the shoulder is badly deformed or the arm becomes cold, pale, blue, or numb.
These signs may indicate damage to blood vessels or nerves.
Call emergency services for shoulder pain with chest pressure, severe breathlessness, sweating, faintness, or pain spreading from the chest to the jaw or arm.
Seek prompt medical assessment after an injury if you suddenly cannot lift the arm, develop marked weakness, or think the shoulder has dislocated.
These symptoms may indicate a significant tendon injury, fracture, or joint injury.
Seek urgent medical care if the shoulder becomes hot, red, and very swollen, especially with fever or feeling unwell.
This can be a sign of infection rather than an uncomplicated tendon tear.
Arrange a routine appointment if shoulder pain lasts for several weeks, repeatedly disturbs sleep, or limits work and daily activities.
Assessment is also appropriate if you notice ongoing weakness, reduced movement, or symptoms that are not improving with simple measures.
The article also says
Shoulder or upper-arm pain can occasionally come from the heart rather than the rotator cuff.
Every sign above is quoted from the sofidex article on Torn rotator cuff. If you are not sure, err upward — it is always reasonable to be checked.