A bulb is connected to a battery having a potential difference of 6 V A voltmeter is connected across the bulb What is the reading on the voltmeter when the bulb burns out?

Assuming you're referring to a simple incandescent bulb . . .

When the bulb burns out, the filament is open, and it is electrically no longer there.
The voltmeter is then across the battery, which is open circuited, i.e. without a load.
The meter measures the open-circuit voltage of the battery, nominally 6 V.