I think there's a 'TRASH' your presents option - can't check because my Tama's asleep...
Try looking in Presents/Friends and choose Presents, select a gift, but instead of wrapping it, I think you can opt to trash it - I nearly trashed some roller blades by mistake once...
Yes tamamum! Go under the book and under presents. Click either food or item and select one. Then it will say wrap? instead of doing that press the "A" button and it will say trash? and select yes.