Both Li and Ca inhibits adenylate cyclase, leading to decreased cAMP, w/o cAMP as 2nd messenger, V2 receptor (Gs) downstream signaling is blocked. This is why. It has nothing to do w/ IP3 mechanism.
Also HCTZ is used for non-Li-induced NDI, amiloride is used for Li-induced NDI, the rationale is that since amiloride blocks ENaC in DT & CD, there is higher Na+ in ductal lumen, and b/c Na & Li compete for reaborption, high Na+ competitive inhibits Li reaborption. HCTZ is not used b/c it increase distal Na delivery resulting in higher than normal Na reabsorption in DT & CD, this facilitates Li reaborption.
To answer the paradoxical question why HCTZ is used for NDI even though it causes hypercalcemia, it is really simple. HCTZ induces production of hyperosmotic urine, remember it acts on the diluting segment, so by preventing polyuira, HCTZ is acting in a way to conserve free water though not directly on the CD.