I couldn’t find a rule on this. We had a couple issues where a magical item came across some monsters that could destroy the material they were made out of (metal, wood, etc.).
I couldn’t find the rule on this, so I wanted to clarify.
Do magical items get any sort of save to avoid being destroyed when damaged, dissolved or broken?
Yes. They use the wielder’s saving throw.
What about for situations where a save isn’t typically allowed, such as a Black Pudding eating a metal sword?
The character doesn’t get a save. They just take damage (after a successful monster attack roll).
Would the item still get saves? Would that be a Save v. Death?
In general, I’d use the rules for gray ooze for most circumstances: Non-magical weapons, armor, and clothing do not receive a saving throw. Magical weapons, armor, and clothing are allowed a saving throw using the wearer’s save versus Death, adding any magical bonus to the roll if applicable.
If the spell or creature specifies otherwise, use the specified rule. For example, rust monsters work differently.
What about unattended magic items?
Maybe the last person to wield it? 
Alex, I would maybe put a note about magical item saving throws somewhere in the magical item section for future editions.
Who leaves magical items unattended?!
It’s the owner’s saving throw in that case.
The over encumbered dude running from the ooze? I’m just guessin’.
“Who leaves magical items unattended?!” Ha. The DM does, quite regularly. Or at least unknown and unattended by the monster(s) “guarding” it.
And then the Party comes along and blasts the room with a Fireball or burns the place to a crisp with military oil to kill off all the pesky ogres and orcs only to find out later that the old shield hanging on the wall was actually magical.
Easy enough to house rule in any case.
I figured the original owner was already eaten by the ooze.
"I figured the original owner was already eaten by the ooze. "
I was the original owner of said magic item referenced in Michael’s original post, and as a matter of fact I was already eaten by the black pudding. Who knew those suckers could move at a full charge straight up a shaft? 
Fayne got restored with barely a scratch. Quit whining.