No, there is no general rule that states that you may make additional saving throws to shake off ongoing effects. Either the spell description, or the description for the specific conditions it applies, will specify if you have the opportunity to make further saving throws against the effect. Otherwise, assume it lasts for the duration.
Many particularly incapacitating spells will feature this, but it is always explicit, not implied.
RAW, no.
Or, answering the body question instead, the spell ignores these features.
Spells do (only) what they say. The spell does not force a saving throw. For the monster's feature to be a "Specific beats general" case, it should explicitly state something along the lines of:
If this creature is affected by the sleep spell, it gets a X saving throw. The consequences for failing are Y. The consequences for succeeding are Z.
Neither of them do, and, as far as I'm aware, no creature or class has a feature worded like that, i.e., no feature states that "they get a saving throw" in any way. They simply have advantage against saving throws that already exist, which sleep doesn't provide.
A similar case is
The mirror image spell has no effect on magic missile, which doesn't involve an attack.
So, we could answer with the same logic:
The bugbear chief's feature Heart of Hruggek has no effect on sleep, which doesn't involve a saving throw.
Additional conjecture for my answer is that you would have to adjudicate what type of saving throw? Is it Wis? Con? What happens if the creature succeeds the saving throw? Is it simply not affected? Does the HP from sleep's HP pool that would have been used for that monster get reduced anyway? While these answers might be easy to adjudicate, none of them is answered by neither the spell description or the feature. Spells are (supposed to be) clear (that's the whole point of "Spells do what they say"), so, if you have to adjudicate so many things, this is probably not how the effect should be resolved.
As for "Why does the Bugbear Chief have a feature like that then?", I've created this question. NautArch has already provided examples. One of them, the symbol spell, has a possible effect that states:
Sleep. Each target must make a Wisdom saving throw and falls unconscious for 10 minutes on a failed save. A creature awakens if it takes damage or if someone uses an action to shake or slap it awake.
Best Answer
The ghast would not get advantage against Control Undead
The ghast has the Turning Defiance feature:
However, Control Undead, although it is a Channel Divinity power like Turn Undead, does not turn undead. Therefore, the ghast does not get advantage against Control Undead as per RAW. Features that turn undead will say so in their description, such as a cleric's Turn Undead feature or Devotion paladin's Turn the Unholy feature, both of which say "If the creature fails its saving throw, it is turned for 1 minute or until it takes damage."
No advantage for attempting to control an already controlled undead
The Oathbreaker archetype (DMG, p. 97) including this on the Control Undead feature:
Although it says that the undead must obey the paladin's commands, a saving throw isn't really a command, so you wouldn't necessarily be able to "order" the undead to fail it (see Can you choose to fail a saving throw? in short, Jeremy Crawford tweeted "No rule lets you opt to fail a save. As DM, I might allow it, assuming you aren't incapacitated or dominated.")
Furthermore, to use Control Undead again on the same undead ends the control you already have over the undead. What that means is you have control (assuming we're still within the 24 hours since the first time you used it) right up until you use Control Undead again, then as you use it, in that instant, you do not control the undead, so it would not be affected by the previous use of Control Undead.