How dangerous is a tarantula?

The New World tarantulas that are found in the Americas tend to have very mild venom and are relatively docile. These are the most common tarantulas found in the pet trade. Most will flick urticating hairs when threatened which can irritate the skin, nose and lungs and damage eyes. 

The Old World tarantulas come from Europe, Asia and Africa. These tend to have much more potent venom and are more aggressive. They are still seldom deadly but can cause serious illness. 

First, there is no such thing as a deadly spider.
Only a few species of the 40,000 species is considered 'medically significant'. meaning a lot of people bitten by those may need medical treatment.
Tarantulas are NOT medically significant, but like all spiders, they are venomous like bees and ants. I thing of their bite like getting stuck by a rose thorn, you will feel it.
You usually have to provoke a tarantula to bite, like grabbing them in the hand, they are very reluctant to bite humans or animals bigger than them.
Most Ts are the new world sort and rather than fangs for defense, they flick urticating hairs from their rump for defense.
In the eyes, nose, or mouth, these hairs are very uncomfortable, even painful. A couple videos for you:
https://www.youtube.com/watch?v=Zu7xpq7uoh0
https://www.youtube.com/watch?v=34XIiXmSq6I