Sounds like it might have been a little too rough and you may have gotten cut. Have you been feeling better since you posted? It should heal, but if it continues to get worse, just have your doctor take a look- it's important to make sure that hands are washed and your partner is gentle- if it hurts, tell them. Hope you are feeling better!
I don't think this is really 'normal' but happens. That area is very vascular. There are blood vessels on places like your cervix which can get irritated and bleed. Or yes, a cut. So, the burning . . . is this while you are peeing like from inside of you around where the pee comes out or the urethra or is it when the pee hits the area. Those are two different things. When we have a cut or something like that, urine will make it sting when it hits it. But if we have something like a urinary tract infection, it burns when we are actually peeing especially as the stream comes to an end. We often have an urge to go more frequently too. Which one is it more like? If it is stinging when you urinate from pee hitting it, then you can get a squirt bottle with room temp water and squirt it as you go to the bathroom on the area. It helps a lot. This is what they have new mama's due after having a baby when they have vaginal tears. Things when they heal also can begin to itch, so that may occur also if this is the case of vaginal tears. If it hurts AS you pee from the inside, then you have to see a doctor as that is possibly a UTI and requires antibiotics. That can happen if fecal matter is transferred from your anus during fingering accidentally to your urethra. It easily happens and something to be careful of.
And agree to tell your partner not to be rough. Sometimes they have no idea and you aren't supposed to have issues after what is supposed to be a feel good experience. good luck