“I drink it for the taste.”
For the taste means in order to enjoy the flavor. Frank is explaining why he drinks coffee. His reason differs from Tina's, who wants another cup to stay awake and keep working. It is not a statement that caffeine has no effect on him.
“a decaf” / “a five-minute break”
Decaf is short for decaffeinated and means coffee with the caffeine removed. A five-minute break is a rest lasting five minutes. Since Tina does not want to lose time, Frank suggests a short break, and Tina accepts because it is that short.
“Too bad the decaf won't write my paper either.”
Too bad ... means "It's a pity that ...," and either in a negative sentence means "also not." Picking up Frank's line 6, that coffee won't write her paper, Tina grumbles that decaf won't write it either. The punchline is that she agrees to the break but jokes that she still wishes a drink would do the work for her.