I've not heard of this as a specific symptom of hypothyroidism, however, I'd never rule it out. If you have results of recent thyroid tests, please post them, along with the corresponding lab reference ranges, since ranges vary from lab to lab and have to come from your own report.
Some people have reported a type of mouth thrush, which is, typically, characterized by a white layer over the tongue. This can be caused by yeast overgrowth, which is not uncommon with hypothyroidism.
In addition, sinus issues/allergies can cause redness of the eyes, irritation in the throat, etc.
It's very important that you take your thyroid medication regularly and properly to get the best absorption from it. It should be taken first thing in the morning with water and nothing to eat/drink for 30-60 minutes...
Excerpt from Wikipedia - Burning Mouth Syndrome...
"Burning mouth syndrome (BMS) is a burning sensation in the mouth with no underlying dental or medical cause."
"Apart from BMS itself, a full list of causes of an oral burning sensation is given below:
Deficiency of iron, folic acid or various B vitamins (glossitis e.g. due to anemia), or zinc
Neuropathy, e.g. following damage to the chorda tympani nerve.
Medications ("scalded mouth syndrome", unrelated to BMS) - protease inhibitors and angiotensin-converting-enzyme inhibitors (e.g. captopril).
Type 2 diabetes
True xerostomia, caused by hyposalivation e.g. Sjögren's syndrome
Parafunctional activity, e.g. nocturnal bruxism or a tongue thrusting habit.
Restriction of the tongue by poorly constructed dentures.
Herpetic infection (herpes simplex virus).
Allergies and contact sensitivities to foods, metals, and other substances (see table).
Human immunodeficiency virus.