No, while you may observe that the app will continue downloading charts and data while in the background, it will only be for a little while. The iOS will eventually stop the downloading.
Downloads need to be completed with ForeFlight open and with the iPad or iPhone connected to the internet. We recommend that you turn on Automatic Downloads so that the app will start downloading updates anytime the app is open and you have an internet connection. To turn on Automatic Downloads, go to MORE > SETTINGS and turn the switch on next to AUTOMATIC DOWNLOADS.
If it’s essential that your downloads finish without interruption, we recommend you leave ForeFlight in the foreground and keep your device awake to ensure that the download proceeds. To prevent your device from sleeping and stopping the download, open ForeFlight and go to MORE > SETTINGS. Turn the switch off next to ALLOW DEVICE TO SLEEP. This action will keep your device awake as long as ForeFlight is in the foreground.