It's true that removing nonessential information, which exists between a pair of commas (as here), should have no significant impact on the sentence:
It's true that removing nonessential information should have no significant impact on the sentence.
However, it's not true that simply having two commas in a sentence means that the information between those commas is nonessential:
I like mangoes, cherries, and kumquats.
Here, although located between two commas, cherries is not nonessential information. You have to analyze the sentence and determine what functional role the commas play.
If something really is nonessential, then removing it shouldn't impact the surrounding text. If things are constructed in such a way that it would, then either it's not actually nonessential or the sentence has been poorly constructed.
This also applies to other punctuation marks, such as parentheses and em dashes. However, there may be some stylistic issues:
What I want to say, and I'm sure you'll agree—in principle—is that sentences should be formatted so that the removal of nonessential information doesn't harm anything.
The problem with this is that I have a single comma and two em dashes. Technically, I should insert a second comma after the second em dash:
What I want to say, and I'm sure you'll agree—in principle—, is that . . .
But although there are now appropriately nested pairs of punctuation marks, this particular use is too awkward.
One solution is to replace the em dashes with parentheses, thereby allowing a second comma without difficulty:
What I want to say, and I'm sure you'll agree (in principle), is that . . .
I might be proven wrong by comments or other answers, but I can't think of a situation where a sentence should be constructed in such a way that nonessential information (and its punctuation marks) couldn't be removed without doing harm to the remaining text.