According to traditional grammar, prepositions have objects. That's a nice simple rule which, unfortunately, doesn't accurately describe the way that the English language works.
Some more modern analyses of English grammar present a different picture. Much like verbs, prepositions can license several types of arguments, and sometimes don't bother to license an argument at all.
The way that you label the word "as" depends on what analysis you understand, on what framework you use, on the way that you define these labels.
If you define preposition to mean a word that must have an object, then "as" cannot be a preposition. You'd need to use some other label, perhaps conjunctive adverb. If you use a more modern definition, then this "as" is a preposition with some kind of argument other than an object. It's a preposition with an adjectival complement.
None of this background information directly addresses the question. Let's do that now:
I hope you won't take my criticism as offensive.
The question is whether this sentence works without the "as". The answer is no. The reason is the verb "take".
Several other good sentences exist that don't need this "as":
I hope you won't consider my criticism offensive.
I hope you won't declare my criticism offensive.
I hope you won't find my criticism offensive.
I hope you won't judge my criticism offensive.
I hope you won't label my criticism offensive.
These verbs are complexly transitive. They license both a direct object and an object complement. The object complement can be an adjective as easily as it can be a noun.
The verb "take" is also a transitive verb, but it is not a complex-transitive verb. It can (rarely) have two arguments. However, when it does, both of those arguments are objects and the verb is ditransitive:
If you're meeting John later, remember to take him his coat. He forgot it this morning.
It's much more common to see "take" used in a monotransitive construction, with a direct object as its one and only argument.
Because the verb "take" does not by itself license an object complement, the sentence in question needs some other word to govern the adjective "offensive". We need something that can attach this adjective to the verb "take". No matter what label we use for the word "as", that word can do that job.