Chris is absolutely right about not overdoing it - I think my hens are getting a bit portly now as I can't resist giving them a bit of something I know they will like (the leftover rice etc) every time I go into the run for some chicken time!
If you treat sweet chestnuts, in their cases, as seed then they should keep though. Keep them outside in a cool, dry place as if you were storing them for planting. Actually. For planting they would be stored in moist sharp sand in breathable bags and exposed to the frosts to break their dormancy to germinate in spring but as long as they are cool and can breath in natural outdoor air they should be fine.
If I had a bucket full I would smash a few with a hammer from time to time and throw them down as scratch. I think the general principle is not to introduce more than 5% of healthy 'treats' into the daily balanced feed.