Afrin - What thyroid disease is referred to in instructions?

No, Synthroid shouldn't be affected by Afrin.

The problem is if you have hyperthyroidism..

Topical Sympathomimetics (Includes Afrin)
Applies to: Cardiovascular Disease, Cerebrovascular Insufficiency, Hyperthyroidism

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Topically applied sympathomimetic agents are systemically absorbed, with the potential for producing clinically significant systemic effects, particularly during prolonged or indiscriminate use. In cardiac tissues, these agents may produce positive chronotropic and inotropic effects via stimulation of beta-1 adrenergic receptors. Cardiac output, oxygen consumption, and the work of the heart may be increased. In the peripheral vasculature, vasoconstriction may occur via stimulation of alpha-1 adrenergic receptors. Palpitations, tachycardia, arrhythmia, hypertension, reflex bradycardia, and coronary occlusion have been reported rarely during the use of ophthalmic and nasal sympathomimetic agents, but may be more likely if the corneal epithelium is damaged or if an excessive amount of drug is swallowed during nasal administration. Therapy with topical sympathomimetic agents should be administered cautiously in patients with corneal abrasion, sensitivity to sympathomimetic amines, hyperthyroidism, or underlying cardiovascular or cerebrovascular disorders, especially coronary insufficiency, cardiac arrhythmia, or hypertension.

http://www.drugs.com/disease-interactions/oxymetazoline-nasal,afrin.html