Will the use of Latisse make my permanent eyeliner fade a lot faster?

No, I don't think latisse will fade you permanent eyeliner. In fact, it might even make it look darker. Be sure to follow the package directions and watch your lashes get longer and thicker. Best, Dr. Nazarian

Use of Latisse should not affect your permanent eyeliner.  Latisse can produce slight discoloration of the skin so you're eyeliner may even appear more pronounced.

Latisse is a prostaglandin analog and although the exact mechanism of action is unknown, it is believed to increase the percent of eyelashes that are in the growth phase and prolong the growth phase. This mechanism should have no bearing on the longevity of permanent eyeliner. 

I have never seen latisse worsen eyeliner.  If anything it cause cause hyperpigmentation, rare, but it can happen.  Best, Dr. Emer.

No, Latisse will only affect the growth of your lashes. Because your lashes will increase in volume this may create a darker effect to the lash line but it will not manipulate or change any other permanent darkening that you may already have.

No, I have not heard of this happening.  Latisse only affects the growth of the eyelash.  It does not affect the results of the permanent eyeliner.  It, however, can cause eyelid skin darkening that may or may not be reversible. It may also cause an increased brown pigmentation of the colored part of the eye which my be permanent.