The average lifetime is the same as the area under the survival curve.
With a Kaplan-Meier estimate (a non-parametric estimate of 1-F rather than of F), for example, the estimate of the survival curve will only hit zero when/if there's at least one death after the last censoring time.
Typically if you have censored data, there's censored observations that last beyond your last recorded death. In that case the survival curve never reaches 0 and you don't have a bound on the mean lifetime.
This is why you can't generally get expected lifetime from a Kaplan-Meier. [You can compute an expected lifetime within some time interval -- so you could compute expected lifetime in the study period for example and some packages will provide that or something similar.]
With a parametric survival curve, it will eventually go to zero; while it's possible to have a survival curve that doesn't have a finite expectation, for many typical choices it will have a finite expectation.