Can I use a silver nitrate stick to burn off an aphthous ulcer?

Resolves on its own. Most apthous ulcers resolve on their own. As annoying as they are, it's always best if a problem can be solved by your own body.
No. It's not a good idea to burn an ulcer. The best way to deal with it is to stop your mouth from being a source of inflammation. I tell my patients to use antacid liquid rinses after each meal and before bed and to stay away from acids in general.
Not recommended. There are several prescription ointments for canker sores that sooth and heal. There is also an ointment called kenalog (triamcinolone) in orabase that acts like an oral bandage to soothe and protect. You should consult your dentist. The cause and frequency is very important. Need a good history and may need a culture or biopsy if it won't resolve.

Related Questions

How do I apply liquid form of silver nitrate to a mouth ulcer on the tip of my tongue?

Mouth ulcer. The answer--you DON"T! Self treatment with corrosives and other ablative medications can result in tissue damage where it wasn't intended.