How do I safely deep fry a turkey?
The other answers touch on the fact that its the release of water from the turkey that interacts with the oil, causes the oil to overflow, and then ignite the burner.

Generally, this happens pretty shortly after you put the turkey in (due do any moisture on the outside of the bird).

To do it safely don't bank on the fact that you've removed all the water - do two additional things:


Do it outside.  In the event something bad happens, better to lose some grass on the lawn than your house.  If you don't have space outside, you can always go to a public park or such.  It may seem like a pain but turkey frying should not be done inside.
There's one absolutely surefire way to ensure than you don't catch the burner on fire by accident.  Don't have it on!  


Prep your turkey.  Have it dry on the outside and completely thawed.
Heat your oil outside.
Turn the burner off
Lower the turkey into the hot oil
Wait 1-2 minutes or so, most of the initial contact between water and oil should over by now.  
Turn the burner back on and cook the turkey.
Also make sure the burner is off while taking the turkey out, just to prevent any spills from becoming terrible.



@ChrisWalker, below in the comments, also provides a method for making sure you don't put too much oil in the pot and thus cause a spill over.