My partner also gets a burning rash after sex. I don't have a discharge or smell.

You both may have a urinary tract infection, or you may have a yeast infection, but there's all sorts of things it could be, including other venereal diseases. (It could even be nothing, just too much irritation from sex.) Is there unusual redness? Do you see any white stuff that looks like cottage cheese in your panties or around your vagina (use a mirror and open the labia).

As usual in these situations you may have to go to a clinic. Often a nurse can handle this.

Just because you don't have a discharge or smell doesn't mean you don't have a Sexually Transmitted Disease. Especially since your partner has symptoms, it can indicate you have it as well. Take you and your partner to the doctor and get some antibiotics.

Be Safe and Be Happy!

~~ Apple Pro ~~

Yes, it is obviously a sexually transmitted problem since you both have it So I don't even understand the question, you go ASAP to your doctor, and don't have sex again till you know what it is.