Burden TO or ON some one/thing?

The word "burden" has an etymology originating from "to bear", I am thus convinced that "on" would be a more appropriate preposition.

Beside the etymology, I feel like taking a stab at the reason why this sentence is confusing by examining the word "becoming".

The phrase is "becoming a burden _ civilization". If one focus on the partial phrase "burden _ civilization." The preposition "on" may be more obvious, but in the context of the full phrase, one can just as well interpret "...a burden" as an attribute to the verb "becoming". For example, "becoming a man" is different from "becoming a dog" because the object into which the subject has become is specified, and one may understand "becoming" as a somewhat isolated action. For many verbs that do not have their own special preposition, "to" is used to agglutinate a verb to a target. So, if the original sentence is understood as:

"Law becomes a burden, and this action has effect against civilization."

Then the preposition "to" may be plausibly used, but if it is understood as:

"Law becomes some burden, and this burden is specifically a burden on civilization"

Then the preposition "on" would seem appropriate with respect to etymology.

Finally, the preposition "of" may be used in lieu of the other two.

I would say it really depends on the context of how you're saying it.

"A burden to do something"

Would simply mean that it is a rigorous task, and is not something one would choose to do. This could be dishes/laundry as a young child, or even going to work if you of age.

"A burden on someone"

Seems to mean that someone/something has become a bother (to the point that it is irritating seeing that item.) It would be the same as someone constantly being in the way of plans wanting to be made, or even something as simple as a family member having to be taken care of by a loved one.