As of Android 8.1, unfiltered bluetooth scans are blocked when the screen is turned off. While it is surprising for such a dramatic change to be made in a minor release of Android, this is certainly an intended change based on the comments in the commit:
Stop unfiltered BLE scans when the screen goes off.
The workaround is to use a ScanFilter with all scans. The new 8.1 operating system code simply verifies that any scans active when the screen is off have at least one scan filter. If those conditions are met the scan results are delivered as in Android 8.0.x and earlier.
In order to set up such a scan, you must use the APIs introduced in Android 5.0 and create a ScanFilter with each scan. Below is a filter that will find manufacturer advertisements for any device from Apple with manufacturer ID 0x004c (this will include iBeacons):
ScanFilter.Builder builder = new ScanFilter.Builder();
builder.setManufacturerData(0x004c, new byte[] {});
ScanFilter filter = builder.build();
Similarly, if you are interested in GATT Service advertisements (like the kind used with Eddystone beacons) you can search for a GATT Service UUID with a filter like this:
ScanFilter.Builder builder = new ScanFilter.Builder();
String serviceUuidString = "0000feaa-0000-1000-8000-00805f9b34fb";
String serviceUuidMaskString = "FFFFFFFF-FFFF-FFFF-FFFF-FFFFFFFFFFFF";
ParcelUuid parcelUuid = ParcelUuid.fromString(serviceUuidString);
ParcelUuid parcelUuidMask = ParcelUuid.fromString(serviceUuidMaskString);
builder.setServiceUuid(parcelUuid, parcelUuidMask);
ScanFilter filter = builder.build();
If needed, you can add multiple filters to a single scan, and any that match will return results. The only real limitation here is that you must know all of the manufacturer codes or all of the GATT Service UUIDs that you might match up front, at least when scanning with the screen off.
You start your scan with code like this:
bluetoothAdapter.getBluetoothLeScanner().startScan(filters, settings, scanCallback);
EDIT: It is also possible to do this with an empty ScanFilter that looks like this:
ScanFilter.Builder builder = new ScanFilter.Builder();
ScanFilter filter = builder.build();
If you use such a scan filter, it will match any advertising packet, and still allow detections with the screen off on Android 8.1, effectively giving you the same behavior on Android 8.0.x and earlier.
EDIT 2: On the Galaxy Note 9 with Android 8.1 and perhaps other Samsung devices with 8.1, scans are blocked with the screen off even with an empty scan filter. Scans are allowed with the screen off with a non-empty scan filter as described above.