Can I fly from Turkey to Italy (direct flight) with post office receipt of renewal of permesso di soggiorno; I am not a Turkish citizen? I would like to have a vacation for a week in Turkey, but I am not sure if I can do that.
It’s your nationality/passport which would allow you to enter Turkey and, for this, you can use its electronic visa application system.
For your return, you would be allowed to take a direct flight to Italy, using your expired residence permit, permesso di soggiorno, and the ricevuta, the receipt, confirming that a renewal application is in process.
The IATA (International Air Transport Association), in its Timatic website for check-in agents (www.timaticweb.com), and in a separate clarification, advises:
Residents of Italy with an expired Residence Permit are permitted to re-enter Italy. They must also have a receipt issued by the Italian Post Office or Police Department proving that they have applied for the new residence permit in less than 60 days after the expiration date. They must return to Italy directly from outside the Schengen area
The IATA statement reflects a Directive of the Italian Government’s Interior Ministry which allows one to leave and return to Italy with an expired Permesso di Soggiorno, as long as one has the “ricevuta.”
You may want to take copies of both linked documents to show airline check-in staff or at Italian Immigration. Should you need to prove that your permesso is being processed, you can do that via the Polizia di Stato site (but do check in advance that you can do so, using the electronic device you'll carry with you).
This only affects your return to Italy; you will not be allowed to transit or enter any other Schengen country.