What is 'its movement'?
Otto's Irresistible Dance says:
A dancing creature must use all its movement to dance without leaving its space...
The implications of this depend on a definition of 'its movement'. Does 'its movement' mean movement that it undertakes on its turn, only, or does it mean any movement that it undertakes on any turn? Unfortunately, the rules are not clear about this.
In general, you cannot move when it is not your turn. If something permits you to move when it is not your turn, that thing itself must then spell out the conditions under which you may move, as an example of specific over general.
Personally, I believe that a strict RAW approach suggests that "its movement" is any movement that is undertaken on its turn, as described in 'Breaking up your move' (emphases mine):
You can break up your movement on your turn, using some of your speed before and after your action. For example, if you have a speed of 30 feet, you can move 10 feet, take your action, and then move 20 feet.
I would thus draw a distinction between something like 'dismounting', which costs 'your movement' and so (I would argue) can be done only on your turn, versus 'standing up', which costs 'movement', and so can be done any time you move, whether on your turn or not.
My interpretation that 'its move' is movement that occurs on 'its turn' is not, however, universally acknowledged. Dale M, for example, has written that 'your move' is any movement you make, even forced movement, as he says in the chat discussion for this answer and others. V2Blast in this answer has said that "The definition of "movement" in the rules doesn't change based on whether it's done on your turn or someone else's."
This question explicitly discusses what "your move" means (thank you Exempt-Medic).
If 'its movement' is only on its turn
Otto's irresistible dance uses up all of the movement of the creature on its own turn. However, on the turn that it fails the save against Dissonant Whispers, it does not have its own movement. Rather, the Dissonant Whispers spell provides it with additional movement and then requires that it use that to move away. Since this is forced movement on another turn, not its own movement on its own turn, Otto's does not apply. If the spell had intended for it not to have the capability of moving at any point, it would have said "A dancing creature must use any movement available to dance without leaving its space..."
If 'its movement' occurs any time it moves
Otto's irresistible dance uses up all of the movement of the creature any time it would have movement available. On the turn that it fails the save against Dissonant Whispers, the Dissonant Whispers spell provides it with additional movement and then requires that it use that to move away. However, as soon as it is provided with this additional movement, the movement is used up by Otto's before it can move away, and the creature cannot leave its current space.
