How long do teacup yorkies live?Do they have a lot of health problems?

They should live as long as any other toy breed if they are healthy and properly cared for. Health problems are greater in smaller bred dogs, ie: teacup. Dental disease, collapsing tracheas, and knee problems are big issues, as well as liver shunts and blood sugar issues.
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Yorkie owner

First of all; did you know Teacup Yorkshire Terrier is not an actual breed? When you say Teacup Yorkie I hope you're referring to a small Yorkie. Yorkie's are a member of the toy group.
If you didn't already know that Teacup Yorkie is not the proper thing to call them, read about it at this website: http://www.elvisyorkshireterrier.com/teacup-yorkie-information.htm

Okay, the life span of a healthy Yorkie is 12-15 years. Health issues often seen in the Yorkshire Terrier include bronchitis, lymphangiectasia, hepatic lipidosis, cataracts and keratitis sicca. Additionally, injection reactions (inflammation or hair loss at the site of an injection) are common. Yorkies often have a delicate digestive system, with vomiting or diarrhea resulting from consumption of foods outside of a regular diet. The relatively small size of the Yorkshire Terrier means that it usually has a poor tolerance for anesthesia. Additionally, a toy dog such as the Yorkie is more likely to be injured by falls, other dogs and owner clumsiness. Due to their small size, Yorkies may be endangered if kept in the house with an undiscerning or abusive person, especially a child.

Read more about Yorkshire Terriers here: http://www.staryorkie.com/yt.html
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Hope this helps. =]