Actually, the answer to this question is a very simple one. Nail vanishes make it hard for the surgeon to follow your progress and keep an eye on how well you are doing during the surgical procedures. Since it is not possible for you to communicate the way you feel to your surgeon during the surgery, it is important for him to read the signs your body is giving to enable him know the way you feel. Whenever you are undergoing a surgery, a probe is placed at your finger tip to enable the surgeon read the level of oxygen in your blood. When there is any kind of artificial polish on your nails, it becomes hard for this probe to read through the nail polish. Additionally, the human nails are known to turn blue following any drop in oxygen level, this color change alerts the surgeon to take necessary actions, but with a polish on your nails, this becomes difficult because he won't be able to notice the color change.
Also, except the nail polish, you are also not supposed to wear any kind of make up during surgeries. Your true skin color can be masked by facial make-up. Your body signals the surgeon how you are coping with the surgical processes, but with the make-ups on, things become hard for the surgeon to notice. Additionally, the surgeon may need to put a tape on your face or any other body part, and with your make-up, it will be difficult for the tape to stick in place.
Other important points to note include leaving every kind of jewelry at home before going to the clinic. Doing this before leaving your home will help keep your jewelries safe from loss. All your jewelries must be removed before the surgery begins, most especially, your lip, ear, nose, belly button, and genitals rings. This will go a long way to reduce the number of bacteria that gain access into the hospital. I just list them here for your information, though it seems not related to the question itself.