Interfacing Atmega 2560 with 74ls240 inverting buffer
You need to connect all grounds: the Arduino's GND and the driver's GND. In the first place this has nothing to do with "feeding" in the sense of providing power, it is necessary to define a common voltage reference. You surely know that voltages (and digital levels are voltages) are measured against a reference, don't you?

If you power the driver and in that way the LED with the 5V output of your Arduino, this will actually make the ground connection additionally a "feeding" power connection.



Note because of @PeterSmith's comment: He's right about the output current for a high state output.

You will measure "high" without the LEDs connected because the driver has no load, beside your measuring device. If you connect the LEDs the load will try to consume about (5V - 1.5V) / 220 Ohm = 16 mA which the driver can't provide. The output voltage will go down until the current is reached that the driver can provide. This will not be the mentioned 3mA because this is the guaranteed value. It will be higher but might be not high enough to make the LED light up.

You might like to consider using a common anode segment LED or the 74HC240 / 74HCT240. The CMOS drivers have a higher current specification.