The short answer here is no, you should not turn off your PlayStation 3 whilst downloading a game. However, please read the information below before giving up on your download, or staying up late and become overtired in order to complete a download.
Most downloads from the PlayStation online store will have a pause/resume function. This allows you to temporarily stop and restart a download for any amount of time as long as the Internet connection is maintained and the machine does not turn off. Therefore you could, for example, pause the download overnight and then return to it the next morning.
You should see a "run in the background" or "download in the background" option whilst you are downloading a game from the PlayStation online store. This allows you to perform other tasks and play other games on your PlayStation whilst the game continues to download and install itself onto your console.
If you purchase and download a game from the official PlayStation online store, Sony will keep a record of this, and you will also be able to access a record of this. In order to access this record, you need to click on XMB, then Account Management, then Transaction History, and finally the Download History option. Once you have navigated to your download history, you should be able to see all of the purchases you have made from the download store. If you do turn off your console during the installation process, or the process is not completed properly, you can download the game again from the download list.
If the above options are not appropriate for you, you can simply leave the console switched on but unattended in order for the download to complete, and return to it the next morning.
- Pause/resume download function
Most downloads from the PlayStation online store will have a pause/resume function. This allows you to temporarily stop and restart a download for any amount of time as long as the Internet connection is maintained and the machine does not turn off. Therefore you could, for example, pause the download overnight and then return to it the next morning.
- "Run in the background" function
You should see a "run in the background" or "download in the background" option whilst you are downloading a game from the PlayStation online store. This allows you to perform other tasks and play other games on your PlayStation whilst the game continues to download and install itself onto your console.
- Use of the download list
If you purchase and download a game from the official PlayStation online store, Sony will keep a record of this, and you will also be able to access a record of this. In order to access this record, you need to click on XMB, then Account Management, then Transaction History, and finally the Download History option. Once you have navigated to your download history, you should be able to see all of the purchases you have made from the download store. If you do turn off your console during the installation process, or the process is not completed properly, you can download the game again from the download list.
- Leave the download alone to complete
If the above options are not appropriate for you, you can simply leave the console switched on but unattended in order for the download to complete, and return to it the next morning.