In my experience, most restaurants (the Italian restaurants in particular in Epcot are very strict) will allow some non-soda drinks. However, non-soda drinks are not refills.
Standard ice tea and this horrific (in my opinion) Pomegranate Lemonade Punch are counted as soda, and you'll get full refills on them. I can't stand the second, but it is there and it exists. Sadly, the good lemonaid they have (listed in the Big Brown Book as a non-alcoholic option at bars) is seemingly never an option.
Most places will then allow one coffee specialty drink, or one juice. My mother got frozen cappuccino's a lot while we were in Disney. I drank a lot of pineapple juice, as it's not something I get often at home. You'll only get one of those, no refills, but they exist.
Of the places you mentioned, Sanaa has a pretty decent hot tea menu (with loose leaf teas) (other places that have good hot tea menus are Brown Derby, basically anywhere at the Grand Floridian and a good amount of signatures though it's often on the dessert menu). Almost any place has Twinings, but Sanaa has a large variety of teas, and you should be able to get a pot of hot tea (loose leaf) on the dining plan. I do not think the mango lassi's are included, though you can always ask.
TREX, not being a Disney owned restaurant, may allow some sort of smoothie or virgin drink (a virgin pina colada, for example)- I've heard people saying they've gotten them, and I've heard people saying they haven't. I don't think BoG or Coral Reef has anything out of the usual.
Breakfast spots often allow you to get a full press pot of coffee (if they have it) as a breakfast drink item on the dining plan, which is pretty cool.
