More specifically, consider the Gray Ooze's ability to...
Corrode Metal. Any nonmagical weapon made of metal that hits the ooze corrodes. After dealing damage, the weapon takes a permanent and cumulative -1 penalty to damage rolls.
If my Forge cleric casts searing smite on his otherwise normal warhammer and I land a blow, is my hammer still gonna get jacked up?
Searing Smite
The next time you hit a creature with a melee weapon attack during the spell's duration, your weapon flares with white-hot intensity, and the attack deals and extra 1d6 fire damage to the target and causes the target to ignite in flames.—PHB, page 274
In 5th edition, Spells only do what they say they do. At no point does Searing Smite say something to the effect of "this spell makes your weapon magical" or "this spell makes your weapon immune to damage" or any other similar effect. Using Searing Smite would not affect a weapon's vulnerability to Gray Ooze's feature, or other similar features.
