As an Oathbreaker Paladin/Warlock, my character gets darkness as an oath spell at level 5, meaning he can cast it using his holy symbol. In the event that he loses his holy symbol, does he have to have wasted a known spell slot to be able to cast it off of his arcane focus?
Yes, as known spells are handled individually when multiclassing
Each spell you know and prepare is associated with one of your classes, and you use the spellcasting ability of that class when you cast the spell. Similarly, a spellcasting focus, such as a holy symbol, can be used only for the spells from the class associated with that focus.
Spell slots are shared between classes with the Spellcasting feature, and Pact Magic is also handled the same way. For Spellcasters with the Spellcasting Feature:
Spell Slots. You determine your available spell slots by adding together all your levels in the bard, cleric, druid, sorcerer, and wizard classes, and half your levels (rounded down) in the paladin and ranger classes. Use this total to determine your spell slots by consulting the Multiclass Spellcaster table.
The Multiclass Spellcaster table is found here.
And for those with Pact Magic (Warlocks):
Pact Magic. If you have both the Spellcasting class feature and the Pact Magic class feature from the warlock class, you can use the spell slots you gain from the Pact Magic feature to cast spells you know or have prepared from classes with the Spellcasting class feature, and you can use the spell slots you gain from the Spellcasting class feature to cast warlock spells you know.
For the example you gave in a comment on this answer:
say my character is stripped of his items and in jail. He could break out of his cell, summon his pact weapon and, with the improved pact weapon invocation, cast darkness to cover his escape. Thing is, an oathbreaker paladin already knows Darkness at lvl 5 and can cast it with a holy symbol, so it begged the question "does he have spend a "known spell" from his warlock class, to cast Darkness with an arcane focus, since he already knows it?" Basically, does he have to know it twice or is he able to cast it as a warlock at lvl 3 and higher
If your multiclassed character does not know darkness as a warlock, but does as a Paladin, you are only able to cast it using a holy symbol as a spellcasting focus, but are free to use either Spellcasting slots or Pact Magic slots to do so (assuming that the slots are of the correct spell level or higher).
Alternatively, instead of a holy symbol or arcane focus, you can use a component pouch.
(You can use a spellcasting focus or component pouch to substitute for the exact material components so long as the material components have no cost and aren't consumed.)
Yes, if you insist on using an arcane focus
However, you could just provide the components.
Spells are separate
You cannot use your arcane focus to cast your Paladin spells.
Warlock, PHB p107:
You can use an arcane focus (found in chapter 5) as a spellcasting focus for your warlock spells.
However, you can cast your Paladin spells without a holy symbol, using a Component Pouch or some bat fur and a drop of pitch or piece of coal.