There is a huge misunderstanding about Greece. Majority of mainstream tourists tend to believe that Greece equals Greek Islands. Well, this is not true. Greece includes the Greek islands, but it is a very diverse country with a very interesting mainland too. So although many of the islands might feel "closed down' ( no tourist infrastructure available on winter and on many islands permanent population is scarce), mainland will be "open" and actually winter is high season for many mainland areas.

For something really alternative I suggest to have a look on Paleros, a small sea side town on western Greece. It will be very quiet on winter, and a great opportunity to meet with locals. It is only 20 minutes drivce to Lefkas town, so you may explore one of the most interesting Greek islands off-season, a great experience (Lefkas is connected to mainland on a small bridge, no ferries included). Preveza is an other interesting town 40 minutes drive away from Paleros, and Ancient Nicopolis a very interesting site.
Antonis, most known as Antonio internationally runs a guesthouse there, offering Greek hospitality with some interesting alternative aspects eg macrobiotic diet, yoga sections etc. His mum runs an attached traditional Greek tavern, New Mill and you are made to feel like at home. Antonio organizes many activities and he is a very interesting person and a mine of local knowledge. Antonio's websites and see if he plans anything interesting around Christmas (in cased he is around, usually he heads for Asia on winter) .
http://www.paleros.eu/
http://paleroslife.blogspot.gr/

About volunteering work, I am mostly aware of Animal Welfare Organisations such as
http://www.friendsofanimals-nf.com/volunteers2009.html (Athens-site not recently updated but you may contact and check)
http://animalwelfare.parosweb.com/home.php (Paros Island)
http://naws.on-naxos.com/volunteer.html (Naxos Island)
Archelon is always a very interesting option with many activities and always needs volonteers in case you are in the right location http://www.archelon.gr/index_eng.php
I know that Aegina had a very active wild Animal Sanctuary and used to host volunteers on site, but I can not find on-line info on that : (

A general portal for various projects on volunteering in elix http://www.elix.org.gr/en/
If above options don't appeal ,I think that in winter, you will meet mostly Greek visitors rather than foreign tourists around, and it worth to explore some of the popular sites such as Delphi , Nafplion etc. An other great option is to check destinations not so popular to foreigners but great non the less, such as Zagori region, Pelion region, or Northern Greece. Of course, no matter what you do, save a few days for Athens. It is a very diverse and interesting city with plenty to see, do and experience.

I'm not sure what you are asking. Why would an entire country be closed over Christmas? The islands are summer destinations, and as such are very quiet over the winter, but on the mainland it's business as usual. Museums and archaeological sites will be closed on Dec 25th & 26th and on New Year's Day, but apart from that you will have many opportunities to "learn".

Wow, Thanks Mariha, very informative and a lot to go on. Much appreciated!

After a quick research on the internet, here is a second option :

http://www.corfu-donkeys.com/Volunteering.htm

I am originally from Corfu, being there off season can be a wonderful experience, Corfu is a very green island, with an amazing old town and many nice villages to explore.

here is some info on Corfu : http://www.corfu.gr/web/guest/home

Volunteers are always needed in Athens where there are many possibilities , especially during Christmas, but i am afraid this would require some knowledge of Greek.