The Dream spell states as part of its effect:
You can make the messenger appear monstrous and terrifying to the target. If you do, the messenger can deliver a message of no more than ten words and then the target must make a Wisdom saving throw. On a failed save, echoes of the phantasmal monstrosity spawn a nightmare that lasts the Duration of the target's sleep and prevents the target from gaining any benefit from that rest. In addition, when the target wakes up, it takes 3d6 psychic damage. (PHB, page 236)
If the victim is affected by Dream during a long rest, they don't gain the benefit of the long rest upon waking from their sleep (for example, a cleric won't recover spent spell slots).
Does this effect interact at all with being unable to benefit from more than one long rest in a 24-hour period? Could the victim wake from their Dream'd long rest (gaining no benefits) and immediately attempt to take another long rest to actually gain the benefits?