Certainly do. The Underground stations at LHR are open for business during the operating hours of the Tube, so staff will be in place from about 0530.

However, since you're travelling on a Sunday morning, you may have to adjust your travel time or mode. The first Piccadilly line train from Green Park to LHR departs at 0706, arriving at T4 at 0752 and at T1,2,3 at 0755 (you'll need to alight at Hatton Cross and wait a couple of minutes for the train to T1,2,3). That might be a bit late for you, even if you do check in on line.

There is a District line train from Earl's Court at 0640, arriving in Acton Town at 0654. The Piccadilly line towards T1,2,3 and T5 calls there at 0700 with an arrival at T1,2,3 at 0722 and T5 at 0727. Green Park to Earl's Court....well, you could take a taxi (££) or the number 74 bus from Hyde Park Corner, which calls at Earl's Court station. Alternately, take the number 9 from Piccadilly to Hammersmith and join the District line there.

http://www.tfl.gov.uk/assets/downloa...ict-170509.pdf
http://www.tfl.gov.uk/assets/downloa...ly-1705209.pdf

And if all that sounds like too much of a pain in the neck, consider a car service from your hotel to LHR. Have a look at the rates here as a saloon car for two costs just £24 from W1.