A little. Thyroid problems play a negligible role in obesity.
The American Thyroid Association states:
In general, 5-10 pounds of body weight may be attributable to the thyroid, depending on the severity of the hypothyroidism.
Most of the weight gain directly associated with thyroid issues is negligible. In fact, the average weight loss from treatment was 4.3kg, most of which was attributed to the excretion of excess body water
Pre-existing obesity, a diagnosis of Graves' disease and prior weight loss independently predicted weight gain and weight continued to rise with time. 
(Common phrases used by users for this page include weight gain with hypothyroidism, weight gain myths fir hypithyroidism, hypothyroidism weight gain, hypothryoid and weight gain, average weight gain from hypothyroid, Can Hyperthyroidism Cause Weight Gain)