I'm still in the process of doing this. I'm keeping cards that really stood out from my childhood (the 1987 Topps set really roped me in), but donating most everything else. I found a nonprofit in my city that was interested in them a couple years ago, so I gave one 5k box to them (that was tax deductible).
Another option is
http://commons4kids.org. I sent a large flat-rate box full of cards to them last month (including a Griffey Jr. 89 UD that Net54 member maximus35 sent to me for no compensation - thanks again!). They donate the cards to a variety of organizations (donations to commons4kids are not tax-deductible though).
I've found that finding a local nonprofit can take a little time. I'm going to put together another 5k box soon to donate, and reached out to a local group last night. The email I got in response noted that they wouldn't be able to use them and suggested I do an online search to find an organization that could (gee - where did they think I found their information to send them an email?).
As for the space it creates, I'm someone who prefers empty space to clutter, so even if I'm not planning on using it for anything else, donating and de-cluttering is the way to go.