When someone takes lithium as medication he has side effects of dry mouth and polyuria for treating those symptoms he is given diuretic. how come?

Lithium: Not all diuretics. Loop diuretics such as lasix, result in lowering of sodium levels in the body, hence cause lithium toxicity. Thiazide diuretics such as hctz, (hydrochlorothiazide) cause lithium toxicity as well. Since lithium competes with sodium, it causes lowering of na levels, & here is where na-sparing diuretics are used to combat that.

See below: Most diuretics will only worsen dry mouth and polyuria. There is one medication Amiloride which is a diuretic- it will decrease polyuria and thirst.

Diabetes insipidus: One side effect of lithium is nephrogenic diabetes insipidus. If this was diagnosed correctly, one treatment option is a diuretic.Mechanism is unknown but it is thought that this will "confuse" the kidneys into retaining more fluids thereby alleviating the polyuria.