Lacrimation is affected when facial nerve injury is at -
First, see the major symptoms of facial nerve palsy : -


Loss of lacrimation: - Due to the involvement of greater superficial petrosal nerve.
Loss of stapedial reflex: - Due to the involvement of nerve to stapedius.
Lack of salivation: - Due to chorda tympani.
Loss of taste sensation from Anterior 2/3 of the tongue: - due to chorda tympani.
Paralysis of the muscle of facial expression: - Due to terminal (peripheral) branches.


Now you can make out the site of injury : -
All the 5 symptoms (i to v) are present → Injury is at or proximal to geniculate ganglion (as all the branches of the facial nerve are involved).
There is no loss of lacrimation (greater superficial petrosal nerve is spared) but symptoms (ii) to (v) occur → Injury is distal to geniculate ganglion but proximal to or at the level of the second genu from where the nerve to stapedius arises.
Only symptoms (iii) to (v) are present (greater petrosal and nerve to stapedius are spared) → Injury is distal to the second genu but proximal to the origin of chorda tympani, i.e., Injury is between Second genu and mid portion of verticle segment.
Only (Vth) symptom is present → Injury is distal to the origin of chorda tympani, which may be at the level of the stylomastoid foramen.
Geniculate ganglion
In semicircular canal
At sphenopalatine ganglion
None