Is the immune system suppressed during sleep?
During stress glucocorticoids, which are steroid hormones produced by adrenal glands (such as cortisol and corticosterone), are released into the blood-stream. These hormones have anti-inflammatory effects. You may say that they suppress the immune system.

The circadian rhythm, which controls the sleep-wake cycle, also controls glucocorticoid secretion, thereby reinforcing the sleep-wake responses [1]. Sleep deprivation is a form of stress and it causes an elevation in the levels of glucocorticoids. The effect of sleep deprivation could be replicated by adrenectomy (surgical removal of adrenal glands); there is a transcriptional downregulation of several neuroprotective genes [2]. Via glucocorticoid mediated signalling, sleep deprivation also results in inhibition of adult neurogenesis in the hippocampus [3]. 

Another study says that cytokine (TNFα and IL6) secretion by immune-system cells, are inherently affected by circadian rhythm via a glucocorticoid independent mechanism [4].

Overall, sleep is inversely correlated with reduced inflammation because both the processes are under the control of the circadian clock. Sleep deprivation, as a form of stress, is also correlated with glucocorticoid secretion. It should be understood that sleep doesn't necessarily inhibit glucocorticoid secretion and it would be incorrect to say that reduced inflammatory effects are because of sleep.