The first let's consider when we can and cannot do this shifting at all. Our hypothetical "he" is going to take the toy away, look at it, and then look it up (in a catalogue of toys or similar). Then he's going to look up toward it (perhaps the toy is a helicopter):
He took away my toy.
He looked at my toy.
He looked up my toy.
He looked up toward my toy.
Consider if the word with the verb is acting as a preposition. This is perhaps easier with away (not generally considered a preposition in any case) than up (which can also be a preposition).
The at is a preposition with the classic school-learning explanation that it is about the relationship between his looking and the toy. up toward is also serving this rôle, though it's two prepositions together.
The away and up are not prepositions in this sense, (despite up generally being considered a preposition and serving as such in other expressions*) because it doesn't describe a relation between the taking and the toy. Instead it's a particle that is giving us a new verb "take away" that names a different action than just "take".
In the preposition (at) case we can't shift the noun because then we change what elements the preposition is describing a relationship between (if any), but we can with the other case (away):
He took my toy away.
*He looked my toy at.
He looked my toy up.
*He looked my toy up toward.
Indeed, it's perhaps easiest to see that away and up are not prepositions in the first set of sentences is by considering that very fact that it still works here, while the preposition at and up toward do not. (Again, despite the fact that up is a preposition in other uses.
Now, of those case where we can shift (away and up) when the object is a pronoun we always shift:
*He took away it.
He took it away.
He looked it up.
*He looked up it.
Notably, while the first here seems just plain wrong, the last seems to have a different meaning; because up can also be a preposition, and because we would always shift it with a phrasal verb of this sort it seems that it must be acting as a preposition in this case and so we are led to wonder what shape this toy must be that "looking up it" makes sense.
While always done with pronouns, the freedom with other noun phrases to shift or not shift and the likelihood that we would, is reduced with the length of the phrase:
He took away my really cool remote control helicopter that I got for Christmas and had been playing with every day since.
?He took my really cool remote control helicopter that I got for Christmas and had been playing with every day since away.
Here the second sentence isn't wrong, but it is unusual and unlikely to be used.
*Which can lead to ambiguity: "He looked up my skirt" differs considerably between a sense that can be rewritten as "he looked my skirt up" and a sense that cannot.