Does the brain need more sleep after heavy load? Does the brain require more sleep in the following night, if you have used a lot more of its capacity that day than usually?
After stressful events the brain is energized by various (evil in this case) hormones from the adrenal system like cortisol, epinephrine and norepinephrine. Good luck getting sleep without some medical help after trying to solve some major world, work or personal issues. So while the brain still needs its normal rejuvenation and maybe an extra 30 minutes or hour to deal with the bad stages of sleep but its probably not going to get it. That is to say there are several stages of sleep and the adreno hormones mess with the length and quality of the stages which can be fixed by a longer amount in some situations or just flat out starting over the next night and trying to catch up on the debt over the course of time.

Stress, Arousal, and Sleep
Stress is considered to be an important cause of disrupted sleep and insomnia. However, controlled and experimental studies in rodents indicate that effects of stress on sleep–wake regulation are complex and may strongly depend on the nature of the stressor. While most stressors are associated with at least a brief period of arousal and wakefulness, the subsequent amount and architecture of recovery sleep can vary dramatically across conditions even though classical markers of acute stress such as corticosterone are virtually the same. Sleep after stress appears to be highly influenced by situational variables including whether the stressor was controllable and/or predictable, whether the individual had the possibility to learn and adapt, and by the relative resilience and vulnerability of the individual experiencing stress. There are multiple brain regions and neurochemical systems linking stress and sleep, and the specific balance and interactions between these systems may ultimately determine the alterations in sleep–wake architecture. Factors that appear to play an important role in stress-induced wakefulness and sleep changes include various monominergic neurotransmitters, hypocretins, corticotropin releasing factor, and prolactin. In addition to the brain regions directly involved in stress responses such as the hypothalamus, the locus coeruleus, and the amygdala, differential effects of stressor controllability on behavior and sleep may be mediated by the medial prefrontal cortex. These various brain regions interact and influence each other and in turn affect the activity of sleep–wake controlling centers in the brain. Also, these regions likely play significant roles in memory processes and participate in the way stressful memories may affect arousal and sleep. Finally, stress-induced changes in sleep-architecture may affect sleep-related neuronal plasticity processes and thereby contribute to cognitive dysfunction and psychiatric disorders.
Stress and Neonatal Sleep
Routine circumcision, done without anesthesia in the newborn nursery, was usually followed by prolonged nonrapid eye movement (NREM) sleep. Since this form of sleep has been described as a low point on an arousal continuum, we consider its increase to be consistent with a theory of conservation-withdrawal in response to stressful stimulation.