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 if a hair product is followed by trouble breathing, faintness, or swelling of the lips, tongue, or throat.
These may be signs of a severe allergic reaction.
Seek same-day medical care for a painful or rapidly worsening scalp area with spreading redness, warmth, swelling, pus, or fever.
These can be signs of an infection rather than simple oiliness.
Arrange a routine appointment if it persists despite reasonable hair-care changes, suddenly becomes much worse, or comes with ongoing itching, scaling, acne, sores, or hair loss.
You should also seek advice if scalp symptoms interfere with sleep or daily life.
The article also says
Oily hair alone is not usually urgent.
Every sign above is quoted from the sofidex article on Oily hair. If you are not sure, err upward — it is always reasonable to be checked.