Mine never did, but I can imagine it can happen, since sore throat can cause ear ache. I found an article, toward the end of it you can read about possible treatment options. Below is the link and some quote from the publication. I hope it helps.
Chronic Orofacial Pain: Burning Mouth Syndrome and Other Neuropathic Disorders (Raymond C Tait, McKenzie Ferguson, Christopher M Herndon
J Pain Manag Med. Author manuscript; available in PMC 2017 Jun 19.)
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC54...
"While treatment for BMS patients lacks evidence-based or consensus guidelines, the first step invariably involves ruling out other conditions that might be responsible for the symptom profile. Ill-fitting bite splints and selected acrylic appliances can result in inflammation and discomfort, while correcting the design and/or composition of the appliance can yield symptom relief [10]. In addition, metabolic disorders (e.g., diabetes) and nutritional deficiencies (e.g., Vitamin B12 deficiency) may represent other conditions that can occasion BMS symptoms [14]. Finally, BMS symptoms can be associated with adverse effects of medications. For example, select anti-depressant medications are frequently associated with side effects such as dry mouth, a frequent part of the BMS profile."
"Observational and interventional studies of Clonazepam have evaluated topical administration of the drug (e.g., 0.5 mg clonazepam dissolved in the mouth for three minutes then spit out with remaining saliva up to four times daily), as well as once daily, low-dose systemic use [27–36]. With topical administration, clonazepam’s action has a quick effect in reducing BMS symptoms (within 10 minutes) and is association with minimal adverse effects. Unfortunately, the duration of action of analgesia and alleviation of other BMS symptoms is short (3 hours to 5 hours) [15]."

My burning mouth (after a year and a half) went way completely. I had a decayed wisdom tooth pulled and I started taking liquid vitamins. Also had some physical therapy to work on my jaw pain. And got a lot of prayer. Not sure what did the trick, probably a combo of all of it. My pain did go to my ears but I think that was more the jaw pain/ TMJ. All those muscles and nerves are so close to each other and interconnected. So keep trying things! I was SO depressed by it but it’s totally gone now. I hope you get some relief soon!