Just to get this out of the way
Probably one of the most commonly uttered phrases around here, but it can't be said often enough: A flat belly is made in the kitchen, not in the gym. It doesn't matter how strong your core is; if it's covered in a thick layer of fat, no one will see it.
And ab workouts do NOT burn belly-fat specifically. See: Is spot reduction necessarily a myth?
Pain as a measurement
The two most common and harmless types of exercise-related pains are "the burn" which you seem familiar with, and DOMS (delayed-onset muscle soreness). The latter is the one you get 1-2 days after exercising.
Neither of these should be used as an indicator of the quality of your training. You can have a fantastic workout and not feel either of them, and you can have a god-awful session, and have to suffer both.
The burn
You ask specifically about the burn. It's the one that we feel when we do an exercise with lighter weights, but we throw a lot of reps into it. Usually when the muscle is getting tired, but since the exercise is so easy, we can keep doing it over and over. The burning feeling is caused by lactic acid building up and entering the bloodstream faster than your liver can convert it to glucose, which is also why it stops burning as soon as you stop repping.
Core training fundamentals
If you want to really build your core strength, you shouldn't always go for the "easy exercise, many reps" approach. Your core muscles, like all the other muscles, respond to multiple types of overloading. Repping out is one of them, but you have to also challenge them by making them do heavy things. This can be done as simply as adding weight to an exercise you're already doing, or by finding a variation of the exercises you're already doing, that adds more of a challenge, and is inherently heavier.
Variety
Would you also recommend doing the same core exercise routine or shifting every 2-3 days?
As with any workout regimen, variety is key. A proper program should ideally contain both light exercises, heavy exercises, explosive exercises, and functional exercises. If it doesn't, find a set of programs that collectively fulfill these criteria, and jump between them frequently.
Also, something to look out for, is training in all three planes of movement. Your core does a lot more than just contract your torso. Your core can also move side-to-side, and perform twisting motions. It can crunch your upper torso down towards your legs, and it can lift your legs up towards your torso. It can also do this from the side, from the back, and while twisting. All of which should be trained.
This is why functional training is such a staple. If you find an activity that requires you to manipulate and move your body in various ways without even having to think about it as a primary goal, then you don't have to worry about every combination of light/heavy/explosive with top-down/bottom-up/side-to-side/twisting.
Personally, I'm partial to climbing, be it bouldering, lead climbing, or trad climbing. But I'm sure people in the comments will think of some other good examples too.
Anecdotally...
Someone once told me they had spent the better part of a decade doing a bunch of "core training" in the gym, but as soon as they had to actually put it to use, it was completely useless. He was able to hold a plank or do situps not until he became tired, but until he was bored of it. But ask him to climb a tree, and he was as useless as an untrained person. Point being, functional training needs to be part of a proper core workout regimen.
What good is a "strong" core, if it's only strong within the strict confines of a handful of very specific movements?
