hello and welcome to HCM,
A persistent burnt smell in the nose can be due to many causes, this is generally known as PAROSMIA
or altered smell.
As rightly pointed out, you need to definitely consult an ENT who will examine your nose for obvious causes which can cause this smell alteration. Let this be a checklist for you:
1. Complete diagnostic endoscopic examination and CT evaluation of the nose and paranasal sinuses
2. Complete blood profiling including ruling out chronic infections which can cause smell abnormalities
3. A neurological examination including CT Brain to rule out any lesion in the brain
goes without saying that you need to be tested with blood pressure and sugar levels estimation
at the end of all these tests you may still be told that everything is normal but anyway you can have a peace of mind.
You can meanwhile start an antihistamine medication and also try steam inhalation.
take care and get well soon
ps: hope there is nothing burnt in your home!!
Dr Sriram Nathan