Yes, though it can be interrupted. If Background App Refresh is enabled for ForeFlight in Apple Settings > General > Background App Refresh, the app will continue downloading charts and data you’ve selected as long as it can. However, iOS can stop any background downloads if it decides that the bandwidth could be better used elsewhere, which can happen when there are other apps running in the background or foreground, or when the download is large or taking a long time. Closing as many other apps as you can by double-clicking the home button and swiping apps up off the screen can increase the likelihood that the background download will continue, but not guarantee it.
If it’s important that your downloads finish without interruption, such as when you’re about to leave for a flight, 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 you can disable “Allow Device to Sleep” near the bottom of More > Settings - this will keep your device awake as long as ForeFlight is in the foreground.