I want to know if I can move "ab" anywhere else.
"Ich setze meinen Bleistift auf den Tisch ab."

Could one also say this?:

"Ich setze ab meinen Bleistift auf den Tisch."

Question

I want to know if I can move "ab" anywhere else.
"Ich setze meinen Bleistift auf den Tisch ab."

Could one also say this?:

"Ich setze ab meinen Bleistift auf den Tisch."

Answer

First of all, "absetzen" has several meanings but none of them would work for a pencil. The verb should be "ablegen" (to put down). Secondly, since it answers the question of "Wo?" (Where?) it needs the dative case, so it should be "auf dem Tisch". (If it's supposed to answer the question of "Wohin?" (Where to?, i.e. direction of movement) which needs the accusative, you'd just use "legen": "Ich lege meinen Bleistift auf den Tisch.")
And most importantly, yes, it would be possible to move the prefix, but a) not to where you moved it, b) it would demand a comma, and c) it would need a larger part of a sentence following, e.g. "Ich lege meinen Bleistift ab, auf dem Tisch, auf dem sich bereits die Zeitungen und der Block befinden."

Hope this helps.

Answer

You can say: "Ich setze den Bleistift ab", meaning: "I lift the [the tip of the] pencil from the paper [to Interrupt my writing]". But this has nothing to do with putting it down on the table. That would be: "Ich lege den Bleistift auf den Tisch" (legen + auf + accusative), or: "Ich lege den Bleistift auf dem Tisch ab." (ablegen + auf + Dative). Anyway, if you use "Ich lege ab", the "ab" goes at the very end of the sentence (more precisely: of the main clause).

Answer

No, it is wrong. The construction of the sentence is not right. :)