Can Botox Lift the Upper Lip?

Botox injected into the depressor anguli oris (DAO), a depressor of the lower lip, can raise the commissure of the lower lip, and therefore the corners of the upper lip.
To reduce the lip to nose distance, we often perform a gull wing upper lip lift or corner of the lip lift in appropriate patients.
A facelift also elevated the corner of the lip slightly as well, as can clever augmentation of the lips with any technique.

Injections of the depressor anguli oris muscles along the jawline will cause the outside corners of the mouth to gently drift upward especially with smiling but they will not lift the upper lip per se.

 Yes Botox can help lift the corner of the mouth, but please be careful with any injections around the mouth.  If the balance is thrown off, you could have an unnatural looking smile until the Botox wears off.  Good luck.

Before considering having neurotoxins like Botox or Dysport injected to lift or "flip" the upper lip, it's important to understand what exactly these products do. Botox and Dysport are injectable medications that temporarily prevent the muscle that's being treated from contracting. When it is injected into the orbicularis oris muscle above the lip, it limits movement in the area. This can result in an unnatural smile and appearance when talking, as well as restricted movements such as drinking out of a straw or even eating. 
An alternative to flipping the lip is to have dermal filler injected from above the upper lip. Doing this will look natural and you can still have regular movement when talking, eating, or smiling.

Okay so the corner of your upper lip turns inward. This is an unfortunate part of aging; something that is essential to correct in order to maintain youth. The best way to correct this is by fillers. Botox into the LOWER lip/chin will not help your upper lip. The only way to stop the effect of the orbiculares, which allows you to close your lips, is to Botox it. I can't stress how delicate this procedure is. The muscle must be paralyzed such that it prevents the inward turning but continues to allow you to close your lips.
There are few physicians that do this with Botox. I frankly don't have the nerve to try this as the potential complications, inability to close the lips for several months, is too great. So before you go to your dermatologist see a plastic surgeon for real possibilities of correction without the potential devastating complications.

Botox weakens the pull exerted by muscles by interfering with nerve transmission.  Du to the anatomy of the muscles around the lips, it will not elevate the lips.
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Botox plus filler or laser are best for perioral, "lipstick" lines. Botox into the muscle called the DAO, combined with filler in the chin and along jaw line, can help a drooping mouth corner.

Did your dermatoloigst mean the chin or area above the lip. Injections into the DAO muscle can change the downward appearance of the sides of the mouth, but not the upper lip per se, nor the area you are discussing.
The desiered look you want can be obtained with Botox but can be very difficult to perform with good results. I would see someone who is confident that they can give this result for you. The injections have to be placed just in the rpecise area. If they are injected a little too high up it flattens the lips and makes them unattracive. You can also, if too much is injected or placed in the wrong spot, have trouble pronouncing certain letters and words, especially p's and b's. There is also a chance you might develop a drool.
So it can be done but be careful!