Does the heart no longer function? Does it go into ventricular fibrillation?
Damage to the brainstem - the part of the brain responsible for controlling breathing and the heart - has different effects on the heart and the lungs.
Every heartbeat is not directed by the brain. Instead, the brain has a "throttle" to the heart, by which it can regulate the heart rate up and down, but the actual heartbeat is initiated by special cells in the heart, called pacemaker cells. In the absence of signals from the brainstem, the heart will continue pumping; this is (one of the reasons) why people who need ventilators to breath for them do not also need their heart pumped for them.
Breathing, however, is directly controlled by the brain. Without signals from the brain, breathing will stop completely (the technical term is "respiratory arrest", but that literally just means "breathing-stuff stopped-ness"). Survival time without breathing depends on a number of factors, but at room temperature, without medical intervention (ie. rescue breaths or a ventilator) and barring any special circumstances, is only a few minutes. Permanent brain damage sets in a few minutes later, even if the person can be resuscitated.
Of course, without any breathing, the heart will stop beating due to a lack of oxygen before long. The general progression will be from sinus ("normal") rhythm, to ventricular tachycardia, to ventricular fibrillation, to asystole ("flatlining").
The heart beats are initiated by the pacemaker cells which are located in the heart itself. So absence of neural input will not affect heart function by much. Though heart can function without neural input, it will not be able to respond to stimuli. The human heart rate is regulated beat by beat through the vagus nerve. This would be lost. The sympathetic system would be able to influence heart rate through the adrenal gland. But even this is delayed. Plus till the levels of noradrenaline falls to normal heart rate will be high.
The contractility control also will be lost pretty much.
The respiratory centers are located in the medulla and in the absence of brain stem outflow the person will be under respiratory arrest which is a medical emergency.