re not quite the same. "Anytime" means "at any time". So "at anytime" would mean "at at any time", which doesn't make sense.
Furthermore, "anytime" is usually at the end of a sentence (like in your examples), not the beginning. So you could say
"At any time, you can remove a buddy from the buddy list by dragging it to the Trash". But if you said "Anytime, you can remove a buddy from the buddy list by dragging it to the Trash", it would sound very informal.