Your teeth will ache/hurt. My teacher told me this sounds unnatural. I think my grammar is correct and the thought can be understood. Does it sound weird for native speakers?
Toothache is a rather specific type of pain. Quoting wikipedia as I don't have access to the cited source:
Toothache, also known as dental pain, is pain in the teeth and/or their supporting structures, caused by dental diseases or pain referred to the teeth by non-dental diseases.
For pain caused by disease I would use that toothache. But for other types of pain, it's best to avoid it, so "if you try to chew steel, your teeth will hurt", or better "... you will hurt your teeth".
Specifically for very cold food as in the comment, I would steer clear of toothache.