You can export illustrator information directly to photoshop as is but if you're willing to export content from photoshop to illustrator with transparency you either have to save it in GIF or PNG format.
If you're exporting images from photoshop to illustrator i sugest you export in PNG-24 format because is far more suitable than GIF.
I think there's a "preserve transparency" option somewhere that needs to be checked before exporting so please take notice.
I hope this helps.
Merry Xmas to you and the rest of the users in this forum!
