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. 
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- Thyroid.org: Thyroid and Weight
- Weight gain following treatment of hyperthyroidism . Clin Endocrinol (Oxf). (2001) Dale J, et al.
- Relations of thyroid function to body weight: cross-sectional and longitudinal observations in a community-based sample . Arch Intern Med. (2008) Fox CS, et al.
- Weight loss after therapy of hypothyroidism is mainly caused by excretion of excess body water associated with myxoedema . J Clin Endocrinol Metab. (2011) Karmisholt J, Andersen S, Laurberg P.
- Treatment of paediatric hyperthyroidism but not hypothyroidism has a significant effect on weight . Clin Endocrinol (Oxf). (2010) Crocker MK, Kaplowitz P.