I have permanent lasik-induced dry eye. Are there any treatments that increase tear production?

Yes. Restasis and punctal plugs can increase the amount of tears on the ocular surface. See your eye doctor.
Dry eyes. I agree with the supplements, but if there are eyelid abnormalities these can lead to quicker evaporation and bad tear film distribution. In addition many meds and dietary supplements can decrease tear production.
Use supplements. Lasik can lower tear product or alter tear composition so the tears evaporate more readily. Treatment is by supplementary tears, and punctal occlusion with plugs. Flax seed oil ingestion can help. There is nothing that actually increases tear volume in these circumstances.