Teething can be an unpleasant experience for many children, while some — I have to say lucky — children and their parents go through the process relatively easy. Teething at most cases brings small unpleasantries such as increased biting, drool rash, irritability, a slight increase in bodily temperature, difficulties nursing, and decreased appetite, but these are nothing serious and they tend to pass relatively fast. [1, 2]
What to do if my breastfed baby refuses to nurse due to gum sensitivity?
Besides relieving teething pain with variety of teething remedies, the benefits of breast milk are also associated with reductions in the risks of sudden infant death syndrome (often shortened to SIDS), otitis media, atopic dermatitis, nonspecific gastroenteritis, lower respiratory tract infections, obesity, type 1 and 2 diabetes, and even childhood leukemia. 
Some breastfed infants respond to the teething pain by wanting to nurse more often, and others will refuse to eat when the pain is at its peak. This sudden refusal is known as a „nursing strike“. Some of the reasons for the nursing strike include nasal obstructions, playfulness of an infant, and of course — gum discomfort caused by teething.  If your child has a runny or stuffy nose as a result of teething, nursing may cause breathing difficulties, so it's important to clean a baby's nose regularly.
Many new parents are afraid that emerging teeth will cause biting during breastfeeding, but this shouldn't be something to worry about as it's possible to teach a child not to bite. It's not possible to bite if the baby is nursing properly because the bottom teeth are covered by the tongue. Of course that most babies will try biting as it's a reflex response to the discomfort, but it's perfectly possible to breastfeed a child with any number of teeth. 
Some babies respond to the gum discomfort and teething pain by wanting to nurse more often — I know because my daughter does — as breast milk contains natural painkillers, but some babies will feel aggravated if their jaws are swollen and could refuse to nurse. Please don't dismiss breastfeeding for this reason only, as the baby will nurse when he gets hungry enough, despite the discomfort. If the baby is frustrated, give up and try again later when he gets even hungrier.
What to do if my baby refuses solids while teething?
It's painful to chew the food when teeth are cutting through, so your child is probably going to accept colder and mushy foods better when she's teething. Mushy fruits such as bananas, kiwis or peaches are great to try in those desperate moments when you feel that your little one is hungry but refuses her lunch. Make sure that anything you want to give her is pureed well during really bad teething periods.
As with everything else, there's the other side, the other type of a child, the one who enjoys gnawing on edible items such as carrot or cucumber while teething. Some parents are really keen on teething biscuits, but we never tried them as our daughter's been an easy teether so far. I've heard only good things, so that's a possible solution.
If she is fussy and eats less for a couple of days, just be patient until the tooth cuts through, it will get easier for the baby, but for caregivers as well. If the child doesn't eat well and refuses the healthy things you prepare, but accepts something „not so healthy“, it's fine, let her have her potato wedges or another food item you usually don't let her eat, it's not the end of the world.
Studies have shown that many parents, especially in rural areas, have low knowledge about teething pain and symptoms and often give wrong or unnecessary treatments. Diarrhea, fever, and weight loss are never the symptoms of teething only, and if you notice them it's important to consult your baby's doctor. [9, 10]
Bear in mind that the food sensitivity is only temporary and as long as the baby is not lethargic and has her usual energy, she is not suffering and you shouldn't worry that much. As with everything else, gum discomfort will eventually pass and your precious little one will be her regular self in no time.