Will bad teeth make your ears ring

No

Well, not to disagree with Curly ( but I will) bad teeth can cause various infections and these can cause ear aches, and many other health issues. Normally not "ringing" in the ears but bad teeth can cause very very serious health problems,

Yes, bad teeth have made my ear ring because my eustacian tube got congested. And eventually hurt a lot.
Your eustacian tube runs from your ear to your throat and is there for drainage of normal mucous. If it swells along any part of it (it's narrow), it can't drain down your throat. A tooth infection can cause swelling at the spot where it should be draining. Ergo, ringing.

But this isn't the tooth per say, but a related infection. ;)