I'm currently thinking about doing an Aasimar Paladin / Celestial Warlock sometime in the future. Now, I've been looking at the Divine Smite feature of the Paladin (p. 85, PHB), which states the following (among other things), emphasis mine:
Starting at 2nd level, when you hit a creature with a melee weapon attack, you can expend one paladin spell slot to deal radiant damage to the target, in addition to the weapon’s damage.
The "instructions" on multiclassing class features on page 164 (PHB) state the following, emphasis mine:
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.
Now, the second feature talks specifically about using spell slots for spells, not other features, while the Divine Smite feature specifically talkes about Paladin spell slots.
Therefore, my conclusion so far had been that - as per RAW - one could not use warlock spell slots for Divine Smite (or any other class features, for that matter).
Then I googled, and found a passage in the PHB Errata, that states the following:
Divine Smite (p. 85). You can expend any spell slot, not just a paladin spell slot.
This clears the first issue - you don't specifically need paladin spell slots.
What about the second issue though, with Pact Magic? Would one be able to use warlock spell slots for Divine Smite?