Does anyone know any example of two elliptically fibered toric variety (3-fold) that are birational to each other? 



Birational elliptically fibered variteties
Over $\mathbb{C}$ all toric varieties are rational, hence any two (of the same dimension) are birational. Many examples of toric 3-folds with elliptic fibrations are given in the following paper <https://arxiv.org/pdf/1110.4883.pdf>.


