A warlock casts witch bolt at a target successfully. On the target's turn, the target casts sanctuary to protect themselves. On the warlock's next turn, can the warlock choose to keep targeting the sanctuary user with the same witch bolt (not a new casting) without making the save?
Yes, the warlock can keep the witch bolt's effect on them without making a saving throw.
Sanctuary only has an effect when the warded creature is attacked or targeted, but witch bolt only targets them once — in this scenario, that's before sanctuary was active. The automatic damage does take an action subsequently, but that action is neither an attack, nor calls for actively re-targeting. (If targeting were involved, witch bolt could switch targets for its later damage — but notice that it can't be re-targeted, so no targeting is involved for the automatic damage.)
Therefore the already-targeted creature does not need to be re-targeted or attacked by the caster of witch bolt to deal the automatic damage, and sanctuary won't impose a saving throw.
Allow me to try and shed some contextual light on the whole inclusion of 'target' in the ongoing damage section of the Witch Bolt text, among others.
I'll start with Hunters Mark, which has an initial target of 1 creature.
"Choose a creature you can see.....until the spell ends you deal and extra...damage to the target whenever you..."
Hunters mark only target's the creature once, but has an ongoing effect granted by retaining concentration on the target. What this is NOT saying is that you are targeting the creature separately each time you want it take effect. Granted, Hunter's Mark is not a 'harmful spell' in context to Sanctuary and thus would bypass the need for a save, but the same contextual line 'the target' appears in both entries and thus serves as a prime example.
Let's move on to Witch Bolt, which also has an initial target of 1 creature.
"....energy lances towards a creature within range...make a ranged spell attack.... on a hit the target takes...damage....and on each of your turns you can use your action to deal....damage to the target automatically..."
So what does this mean?
What this is saying is that you target the creature once, as per selecting which creature will serve as 'the target' for intents and purposes of the remainder of the spell description. There is one single targeted harmful attack roll against that target. From then on, there is non targeted automatic damage against the creature you selected when you first cast the spell, which we (the entry) shall start calling 'the target' for purposes of preventing confusion. What this is NOT saying is that 'the target' must be selected each time you choose to do ongoing damage. It's already locked in, the creature has now become 'the target' without having to have been reacquired and the spell doesn't give you a choice to change targets, and thus it simply just works.
Sanctuary can save you from a spell where the caster has to choose you as a target upon the time of casting only, but it cannot and will not save you from ongoing damage that doesn't require you to be selected once more (or at all in the case of AoE) as the target in question.
It looks like (assuming the Warlock fails the save) the answer is 'No'
DnD 5e Basic Rules p. 100
You ward a creature within range against attack. Until the spell ends, any creature who targets the warded creature with an attack or a harmful spell must first make a Wisdom saving throw. On a failed save, the creature must choose a new target or lose the attack or spell. This spell doesn’t protect the warded creature from area effects, such as the explosion of a fireball. If the warded creature makes an attack or casts a spell that affects an enemy creature, this spell ends.
The warlock is not making an additional attack roll, but they are targeting the warded creature. The text for Witch Bolt specifically says that on each turn, you may deal damage to the targeted creature, implying that the creature is targeted by each subsequent Witch Bolt blast (and as such would be protected by the Sanctuary spell).