There are four sinuses attached to your nose. The one that has the most dental significance is called the maxillary sinus.
As shown in the animation, the sinuses are very close to the roots of the upper back molars. This means any problems from teeth can affect your sinus and vice versa.
If you have frequent sinus issues, eg. blocked nose, thick nasal mucous, facial pain, be aware that the pain can mimic a toothache. A change in orientation of your head may change the character of sinus pain. A toothache will not be affected.
A tooth infection can also go into the sinus and could require surgery if severe.
Removal of the back teeth may develop a hole between the sinus and the mouth. An pre-operative x-ray would assist in determining the risk of this happening and your dentist may refer you to a specialist if the risk is deemed to be high. If this does happen, you may find drinks coming out of your nose!
On the topic of extractions, roots may inadvertently get pushed into the sinus. Once it is into the sinus, it is extremely difficult to remove and if left, may cause chronic sinusitis.