Yes when your body is sick your body burns more calories.
The reason your body burns more calories when you are sick is because when you're sick your body uses more energy and when your body uses more energy your body then burns more calories.
Also when you're sick you usually lose your appetite and do not want to eat any food so your calories will be less that you consume.
When that happens then your body uses the energy from your fat to produce the energy to keep it working and thriving.
So you do burn a small amount of calories when you're sick.
Although you likely won't notice much weight loss by the calories that are burnt when you are sick but you do burn small amounts of calories when you're sick.