No not routinely. Now that I've learned not to overfill the jars, especially when canning meat, it's not a big issue. If the water is oily/greasy then yes I dump it, hose it out , and refill. My last canning two months ago I ended up with 85 pints and 24 pint-n-1/2's of chicken using 2 canners in 3 runs plus one more batch in the small one. Luckily I didn't have to rinse and just added a little water before the 3rd run.
Leakage doesn't effect the food in the next batch but it can make a mess on the jars that is a pain to clean up before you put them in the pantry.
We Rinse With Chlorine Bleach Solution After Each Batch ...
We cool then quickly wipe out and clean everything with a dilute bleach solution. About one teaspoon of laundry bleach per one gallon of rinse water. If the pot is dirty, more washing before sanitizing.
Our procedures are above using medically defined clean technique and kinda below using the harder medically defined sterile technique. Face masks used. No breeze. Very clean almost sterile gloves.
So far sosss good. This is kinda one of those things were being slightly overboard is much better than being slightly underboard. Everything is handled like a surgery suite for an operation. Overkill? Dunno.
Death by food poisoning is not an easy way to go. Yikes! No eye, ear or nose picking allowed. Also not coughing, spitting, eating, drinking or passing gas allowed. Still lots of gun to home can stuff. Ex RN.