The strange thing is that the order V Adv DO is idiomatic only when the DO (direct object) is heavy or the Adv is a single particle.
Admittedly, as pointed out in another answer, people do sometimes write phrases such as "turn on or off the tap". Yes, it's used, but, if the DO is light, people are far more likely to put it next to the verb. We are far more likely to "turn the computer off and on" than to "turn off and on the computer".
Yes, some confouding factors exist, but, to avoid them, consider sentences where the DO is a light noun phrase. (The DO can't be just a pronoun; if it is, it must go before the other argument.) And consider the question not whether or not a phrase is attested at all but whether it is idiomatic. On this basis I would exclude "turn on and off the TV".
So, then, what can go in the Adv position even if the DO is light, the resulting pharse being idiomatic?
It's curious. Some adverbs work, such as on, off, up, down, as the OP and others have noted. Other examples:
Don't let in the cold.
Put out the candles.
They work whether the particle has no independent meaning, has a figurative meaning, or bears its usual meaning.
Lock up your valuables.
Turn up the volume.
Lift up the lid.
Typical adverbs used in this way make for unidiomatic phrases:
*I turned sideways the picture.
*I visited recently/yesterday London.
*I read quickly/fast the book.
*I watched excitedly the film.
The fact that each of the particles which work is a homonym of a preposition might suggest that it is a preposition, and that the pattern here is that prepositions work but adverbs don't. One trouble with this is that prepositions can be conjoined, and the result can be used the way a single preposition can:
- I walked up and down the path.
whereas, as has been noted, such phrases as "turn up and down the volume" are rare and unidiomatic. Another is that, with some "prepositions", the word order with DO last doesn't work:
And at least one adverb that does work is not also used as a preposition:
- That brought back some memories.
But I have difficulty in finding where to draw the line. These last examples bring home the difficulty.
I tentatively suggest that there is a lightness constraint on the adverb. A test failed by coordinations such as "off and on", and by most adverbs, but passed by a few adverbs. The latter are short. But the matter of whether or not the word can also be used as a preposition is not a reliable guide to whether or not it works in this context.