But I also don't want to be in the situation that someone remembers their JD account in a year's time, asks me for their coins and I have to say "sorry, I gave them away". Any suggestions? Is it reasonable to give a cut-off date after which remaining balances are forfeit?
Cut-off date is perfectly reasonable. Well, unless you're planning to re-launch the same site from different BTC/gambling friendly country. Then you can just move the investments there, with the same login/password.
But if that's not the case, just give everyone something between 2 weeks - 1 month time, announce that on official threads + /r/Bitcoin and /r/dogecoin and it should be fine. After all, everyone could submit email/emergency withdrawal address for the situations like this one. If they didn't - their loss.
Think you should keep any unclaimed funds just to be on a safe side. There's always a risk that authorities can question you about it (not likely, but still). And just hold on to it for whatever is the expiration period, after which they cannot charge you with anything (5 years?).
PS. I used both JD and DD, it was a pleasant experience. Thanks for that