For physical attacks, the scenario where you don't add your Proficiency bonus to an attack roll is relatively clear-cut: when you don't have proficiency in a weapon. If you're a Sorcerer trying to wield a Rapier, with a DEX bonus of +3, then your Attack Roll is +3; no Proficiency added.

However, for spellcasters, I can't find a single scenario where the Proficiency Bonus does not get added to a spell, either to the Attack Roll or the corresponding Saving Throw DC, except, of course, for spells which don't have an Attack Roll or Saving Throw.

Obviously, I respect the balancing implications of this choice: if, for example, things like Racial feature spells/cantrips didn't scale with a character's Proficiency bonus, they would end up being very weak as a character leveled up, whereas with the proficiency bonus, a Level 20 Fighter casting their racial spell Burning Hands can at least expect to challenge the Saving Throw capabilities of their targets, even if the damage is pretty pitiful relative to the kinds of creatures that would pose a meaningful challenge at their level.

But it is strange to me that there appears to be an entire class of features in the game where there's no variance as to whether a character's Proficiency Bonus ought to be added, or not. So it begs the question:

Does there exist some corner-case scenario in 5th Edition D&D where a character would not add their Proficiency Bonus to the Attack Roll or Saving Throw DC of a spell they cast? Or does such a scenario simply not exist?

I can't provide evidence to confirm a negative, but in my experience (thousands of hours of DMing and playing), such a scenario does not exist. If a character is casting a spell personally rather than via an item, they will use their proficiency bonus. Some items - scrolls and wands are the most common - have a fixed DC; the character activating the item has no effect.

By contrast, lacking the other half of that equation can exist. A Thief Rogue with Use Magic Device does not have a spellcasting ability, and therefore uses +0 for magic items that may require one.

From the Basic Rules, Chapter 14, Activating an Item, Spells:

A magic item, such as certain staffs, may require you to use your own spellcasting ability when you cast a spell from the item. If you have more than one spellcasting ability, you choose which one to use with the item. If you don't have a spellcasting ability - perhaps you're a rogue with the Use Magic Device feature - your spellcasting ability modifier is +0 for the item, and your proficiency bonus does apply.

Spell scrolls have fixed attack/save modifiers

The attack modifier or saving throw DC of a spell cast from a scroll is dependent on the level of the scroll, rather than the spellcasting ability or proficiency bonus of the caster. Looking at the modifiers given for each scroll level, they seem more or less in line with what would be expected from the proficiency bonus + spellcasting modifier for a caster whose highest level spell slots are of that level. So the intent seems to be that a spell scroll is using someone's proficiency bonus, it just isn't using yours.

No such scenario exists for known, prepared, or innate spells.

If a spellcaster is intrinsically capable of casting a spell due to a class or racial feature that lets them know or prepare a spell or cast it innately, their proficiency bonus always applies to their spell attack bonus and spell save DC. For magic items or spell scrolls, there may be a fixed bonus that doesn't benefit from proficiency (such as +7 for a 3rd level spell scroll).

Note that with weapons, if you're proficient you can attack well; if you're not proficient, you can attack poorly, but nonetheless you can attack. With spells, however, either you can cast the spell or you simply can't at all. There's not really such a thing as being able to "sort of" cast a spell, except the scenarios covered in the previous paragraph.

Notably, spells such as counterspell and dispel magic require spellcasting ability checks, which by default no spellcaster is proficient in. However, Abjuration Wizards of 10th level get to add their proficiency bonus even to those checks. This implies that Abjurers are so good at those spells that they have become proficient in executing them better than other wizards, yet other wizards can still execute them to lesser effect. These aren't attacks or saves, but they indicate a niche where proficiency doesn't usually apply but could under certain circumstances.