Can 6 year olds get fevers from a new tooth coming in?

They can if the area where the tooth is coming in is infected. However I'm sure you would have noticed this. Alternatively (and much more likely), having a new tooth come in can make anyone feel slightly run-down (lack of sleep from tooth ache, not wanting to eat etc...) and hence more prone to infections. I suspect that your child has picked up a cold/sore throat virus which is causing the temperature. It's probably best to treat the fever as if it were caused by a general virus - with paracetamol and fluids (or whatever your normal routine is).

If they're old enough, you can try getting them to rinse their mouth with something called "Difflam Oral rinse" - it's an antiseptic mouthwash which is really good at getting rid of sore throats/mouth infections. However I'm not entirely sure how old a child has to be before they can use mouthwash (risk of swallowing it).