I am a big tea drinker. Usually just plain black Lipton for breakfast - then a variety throughout the day. When we go to WDW, I usually take an electric tea kettle and a variety of my favorites for the mornings.
But, here are some of my favorites throughout the World:
AK - Royal Anandapur Tea Co. in Asia (THE BEST

- lots of specialty teas and lattes - you won't be able to choose just one!)
EP - Lotus Blossum Cafe in China - Chinese & flavored teas - very good - hot and cold - also Bubble Tea

Tangierine Cafe in Morocco - Mint Tea - Yummy

Tea Stand in China - Lots of frozen tea slushes, flavored and bottled teas, green tea ice cream

Tea Shop in UK - I always like to go early in the trip so I can buy some teas for mornings in my resort room. (Of course, you might just have to pick up a teapot or cup while you're there!

)
DS - Nothing special, but I know you can get a cup of hot tea at ABC Commissary, Rosie's, BackLot Express and Starring Rolls. If you like Earl Grey they used to have it at Writer's Stop.
MK - Tomorrowland Noodle Station - Lots of Asian hot teas (love the mango!) Also a variety of iced and flavored green teas
Main St. Bakery has hot tea to go with your goodies
ToonTown Farmer's Market - I have gotten a nice cup of tea and a piece of fresh fruit for breakfast - pretty good!
I think most of the table service restaurants offer hot tea. It may not be on the menu, so just ask. If they don't, I often carry tea bags and ask for a cup of hot water.
Beware of the iced tea. Some places have fresh brewed (Yorkshire Co. Fish Shop for one) but many are instant Nestea.

It can be as bad as the coffee. Ask before you order!
HTH! Have a great trip!