I am pretty sure the answer is no, but I have only inked about two pictures so far, and I can't remember doing any burnishing before the color fill, but I thought I would ask anyway!
Generally any time you add something on top of a burnished paper, you are adding another microscopic Himalayas which you need to flatten down to the Netherlands so that you can add the next layer! Layers can be single lines - no need to burnish after every single brushstroke of course, but when you've gone without burnishing for a while then have a feel and if it's no longer smooth, burnish. After inking in and before colour fill you may have lots of tiny hills which need landscaping before applying paint yes! It's also about habit forming, and to do it after the inking is a good practice.
I did the 30-minute burnishing of the paper per VP. For the Fire drawing I decided to use walnut ink. A couple places could use some smoothing out methinks.
Generally any time you add something on top of a burnished paper, you are adding another microscopic Himalayas which you need to flatten down to the Netherlands so that you can add the next layer! Layers can be single lines - no need to burnish after every single brushstroke of course, but when you've gone without burnishing for a while then have a feel and if it's no longer smooth, burnish. After inking in and before colour fill you may have lots of tiny hills which need landscaping before applying paint yes! It's also about habit forming, and to do it after the inking is a good practice.
I received the answer to this question from Vaishali in Lesson 2 of the Fire class. Yes, one should burnish (gently) after the inking.
I usually burnish my paper before tracing/transferring as it will make the paper smoother and you will have more brush control for fine details.
You can burnish after inking and it seems to make the ink less soluble if you are not using indelible ink. But I'm not sure it's required.