Lips Drooping After Restylane

This is most likely the bulk of the product causing the asymmetry and not nerve injury.  I would wait until at least 2 weeks after the injection to see if the problem resolves on it's own.  If it is still a problem, consider just dissolving it all away with an injection with Hyaluronidase (an enzyme used to dissolve Restylane).  The most common complication from Restylane is improper placement of the product and that sounds like the problem in this situation.

As the other panel members suggested, the more likely scenario is that you have an uneven distribution of filler product.  This can be easily remedied with hyaluronidase which will dissolve the product.  Most importantly, it's imperative that you express your concerns to the one that did your treatment for a proper assessment and remedy for the situation.  Your appearance is your most important asset, being proactive will benefit you.

The crooked smile comes from Botox and unlikely from Restylane injections. If you cannot move a part of the lip and no Botox was injected, then you should see a neurologist.
If there is only asymmetry between the two upper halves of the lip , it can be corrected by either injecting more Restylane one one side or removing some from the other using Hyaluronidase.
Regards