Go through these in order and stop at the first one that applies. If you are reading this for someone else, answer for them.
Black hairy tongue itself does not usually cause an emergency.
Call emergency services for trouble breathing, severe difficulty swallowing, drooling with an inability to swallow, or rapid swelling of the tongue, lips, mouth, or throat.
Seek a prompt medical or dental assessment for a mouth lesion that is painful, hard, ulcerated, bleeding, growing, or located mainly on the side or underside of the tongue.
People with a weakened immune system should also seek advice about new or persistent mouth changes.
These symptoms may indicate an allergic reaction or another urgent condition and should not be attributed to hairy tongue.
Arrange a routine appointment with a dentist or doctor if the tongue change is unexplained, does not begin to improve with gentle oral care, repeatedly returns, or causes troublesome taste, breath, burning, or gagging.
Follow-up is also appropriate if dry mouth is ongoing or if a medicine may be contributing.
The article also says
Once the discoloration clears, continued gentle tongue cleaning and management of contributing factors may reduce the chance that it returns.
Every sign above is quoted from the sofidex article on Black hairy tongue. If you are not sure, err upward — it is always reasonable to be checked.