Some items increase the saving throw DC of spells cast using them or while holding them. For example the Rod of the Pact Keeper says:
While holding this rod, you gain a +[1-3] bonus to spell attack rolls and to the saving throw DCs of your Warlock spells.
If a Warlock cast a multi-round spell, such as Hunger of Hadar, while holding a Rod of the Pact Keeper then stowed the rod to wield a weapon would the saving throw DC for the affected creatures drop?
It does say in its description "While you are holding this rod...", so you need to hold it to gain those effects.
The wording here describes a spell (which we can infer means a particular casting of a spell) as having a DC, not the caster themself. The wording in the PHB is similar when it says "Here's how to calculate the DC for your spells:", rather than "Here's how to calculate your spell save DC".
In both cases, the wording favors the DC being a property of the spell, not you; you cast the spell, and it derives its DC from your abilities, the spell itself is granted that DC, it's not constantly checking back if you're still as smart/wise/charismatic as you were when you cast it, nor is it checking if you still hold a particular item. For non-concentration spells, they can continue working even if you're dead (i.e. you're now an object, with no stats at all), and there's no implication in the rules that spells end when you die (in fact, there's a specific statement that merely being Incapacitated renders you unable to voluntarily end a spell).
At the moment you cast a spell, unless a class feature or item says otherwise (there are some effects I've seen that specify they affect only the first save), the save DC is locked in. If you cast without the Rod, drawing it won't make later saves harder, but if you cast with the Rod, stowing it won't make later saves easier (for that matter, changes to your stats or proficiency bonus won't affect it either).
This makes sense narratively as well. I view a spell with an attack roll as being a spell effect that must be aimed by the caster themself (no different from aiming a bow), while a spell with a saving throw does not rely on aiming, the spell's power is embedded in the spell itself and the target resists it, or not, based on the power imbued in the spell when it was cast (there's a weird in-between case for single target Dex saves, where attack rolls imply armor helps, while saves imply it doesn't, but all abstractions are going to leak sometimes). The Rod makes a spell more powerful when cast, and focuses the caster for aiming, but putting it away doesn't reach out and suck power out of a spell that's already existing, possibly entirely independent of the caster.
