No, we don't drink tea from our fingers.
It helps if you put the cup all the way up to your mouth.
Amelia likes to fill the tea pot with ice and water all by herself, and she has a tendency to fill it to the very tippy top. Then she likes to pour it all by herself into the tea cups.
We ended up with two happy girls and a tea tray, tablecloth, table and floor covered in water. :)