Matthew 6:16 Moreover when ye fast, be not, as the hypocrites, of a sad countenance: for they disfigure their faces, that they may appear unto men to fast. Verily I say unto you, They have their reward.
Mat 6:17 But thou, when thou fastest, anoint thine head, and wash thy face;
Mat 6:18 That thou appear not unto men to fast, but unto thy Father which is in secret: and thy Father, which seeth in secret, shall reward thee openly.
So basically, business as usual.
Fasting can bring about some very offensive Body Odor s, i.e. breath, skin, etc. Showering with soap and brushing one's teeth with toothpaste isn't going to do any perceptible harm to the fast. And people won't be put off by your fasting as a result, and you may feel refreshed. Diligent hygiene during the fast may help remove excess toxins. The odors coming from the mouth or skin are toxins and thorough washing with the appropriate hygienic products can better remove the gunk than with water alone.