They are not equivalent. "On the way" means that the location or object is along, adjacent to, or near some route.

If we're going to John's house, can we stop by the store to pick up some snacks? It's on the way.

"In the way" means that the location or object is directly in the path or blocking that route.

You used to be able to see the ocean from that hotel, but recently they built a larger hotel that's in the way.

Another way to think about it is that something "on the way" is convenient, while something "in the way" is an obstruction. It's possible to have both in the same sentence:

On the way to Lauren's house out in the country, we were delayed by a herd of cows that were in the way.

Moreover, "on the way" only really makes sense when talking about something related to people, or other entities that have the ability for conscious choice, even if it's only to specify some point of interest.

The restaurant is on the way to our hotel, if you want to stop for a quick bite.

"Fire" would not qualify, as it is mindless. A fire would not stop at a corner store to pick up a bag of chips (a.k.a. crisps) on its way to burn a town. Instead it might burn down the store, if it was in the way, which is what you should use in this case:

The fire consumed everything in its way.

I can't think of any case where you can freely substitute one for the other without changing the meaning.

(Edit) In the case where you're talking about a conscious entity acting as a destructive force (such as an army), which you use depends on what you want to say. For example, if you want to imply the army was engaged in wanton destruction then, as with fire, "in the way" makes more sense.

During the march south, the Union army indiscriminately torched any towns that were in the way.

On the other hand, if you want to imply the army was consciously selecting targets for destruction, then "on the way" would work:

During the march south, the Union army deliberately torched railways, supply caches, farms, factories, and anything else on the way that might be of strategic value to the Confederates.

They can mean similar things in many cases, but they are definitely not always the same. For instance, 'in' can mean 'inside [of]' while 'on' can mean 'on top [of]'. There are too many use cases to explain. You pretty much have to know them all in a case by case sort of way.

In your example, they mean different things, but the effect could be similar.

'The fire consumed everything in its way' means that the fire burned everything that was blocking its path.

On the other hand, 'The fire consumed everything on its way' literally could mean 'everything' (rather than just everything in its path). However, people would probably understand your intent.

Doing something on the way doesn't have to mean that the direct object is nearby: e.g. 'I called my friend on my way to the dentist.' Your friend could be in another country.

A fire could destroy things far away on its way, too. For instance, the destruction of a bomb factory in its path could blow up the whole town on its way. I think it would be reasonable to say the fire destroyed the town, but people could argue that. Or a fire might cause a beam to fall that triggers a switch that makes a rocket launch that causes who knows what else to happen. Anything the fire could do during its journey could be done on its way.

However, when it comes to humans, there is sometimes a context that implies convenience, as Andrew mentioned (but that context doesn't always exist). For instance, "Since you're already going to Chicago, will you pick up some stuff for me on the way?" That's about convenience, while, "I called my mother on the way" isn't necessarily.

It's pretty common for 'in' and 'on' to mean different things.

They are not identical in meaning, but the outcome MAY be the same.

"on it's way" relates to objects in close proximity to the path that it follows.
Here
- "way" is essentially a reference to a physical route, although most would not intuitively see it as meaning this.
- "on" means "in immediate proximity to", besides.

"in it's way" refers to items BLOCKING it's path or lying ON (rather than next to) the path it took so that the fire passed across or through them.
Here
- "way" is again a reference to the physical path taken.
- "in" means blocking, occupying, lying across.

I would add that "in its way" can be ambiguous. Is there any anthropomorphism of the fire? Fire has a specific way in which it consumes, so you could be saying that the fire consumes everything in that way. Another example of that use would be.

"The man seemed cold and distant, but he loved her, in his way"

The implication of that is that the man did love, but had an odd way of showing it, for whatever reason. He had his ways, his mannerisms, and you had to know him well in order to understand his meaning. If you know fire, you know the way it consumes - flames licking, objects turning black and charred, smoke wafting off. You know fire, and you don't need that stuff explained. Without any context, I could easily see someone personifying fire, describing it as mean, angry, merciless, with mannerisms and destructive mannerisms.

. That's very different from the use mentioned in other answers, such as

"The man kicked the dog, because it was in his way"

Here the man is trying to storm off somewhere, but the dog is right in the path - the interaction would not have taken place if the dog had been off to the side.

"On its way" only has one meaning, as far as I know.

"The man patted the dog on his way to the other room."

The dog is near enough to the path that the man could pat it without veering off, or without feeling like he was going out of his way. The distance is subjective, and highly dependant on circumstance, but the idea is that the dog is neither an obstacle, nor a burden to reach.