If I spill food inside a hot oven that is used for meat and dairy, does the oven need to be kashered?
No need to be kashered. Just clean the spill.
follow up on this:
if food spills in the oven while you are cooking, can you continue cooking or do you have to take the food out and clean up the spill first?
Are you cooking a meat dish and milk dish at the same time?
Assuming you meant one after the other, there is no need to clean it till you are done.
I was cooking a meat dish, and some of the meat spilled in the oven. I continued cooking the meat, and then cleaned up the spill before I cooked dairy in the oven.
Sounds like I did the right thing.
Wonderful