Inheritability of facial features
Let's start with the origin of specific facial features. Below is a table which shows averaged heritabilities for a range of facial quantitative traits from a large number of studies, from the review by Kohn (1991):
where $h^2$ is the narrow-sense heritability. You can clearly see that heritability in the majority of the features is well above 0.5, in fact the overall mean from all the features is 0.72 (with standard deviation 0.18). To put it in the context, this review shows that facial features are typically more heritable than behavioural traits (usually less then 0.50), but less heritable than the height (0.8-0.9). At this point we can conclude that a substantial part of the variability in facial features can be explained by the inheritance (i.e. genetic factors).
Cognitive aspects of physiognomy
As @Xurtio pointed out, physiognomy gained and lost popularity several times in history. In general, modern science is sceptical towards the traditional approach to physiognomy (i.e. specific facial features correlate with objective personality traits). However, there is a strong interest in attribution of personality traits to specific facial features. Hassin and Trope (2000) conducted a number of experiments examining the cognitive aspects of physiognomy. In examining "reading from faces" authors demonstrated that physiognomic information changes the interpretation of verbal information. The more ambiguous this information is, the more perceivers use the face. Furthermore, even when asked to, participants were unable to ignore people's faces while simulating decisions regarding personnel selection, although they are quite sure that they are able to do so (Hassin and Trope, 2000). Finally, physiognomic information makes us highly overconfident about our judgments - our confidence in physiognomy-based judgments far exceeds the actual accuracy of these judgments.
We all know that face is powerful information channel that is used to communicate a broad spectrum of emotions, more or less universally across cultures (Ekman et al., 1971). We also have dedicated brain areas specialised in decoding facial information, for example Face Fusiform Area (FFA) that contributes to facial recognition (Sergent et al., 1992). It is therefore not surprising that we all attribute so much to the specific facial features. And that we are biased in making personality judgements based on facial features, as showed by Hassin and Trope (2000).
Regarding the direct correlation between facial features and objective personality traits, there are no systematic empirical studies that support such links (as for September 2015).
Kohn, L. A. P. (1991) The Role of Genetics in Craniofacial Morphology and Growth.
Annual Review of Anthropology 20, 261-278.
Hassin, R., Trope, Y. (2000) Facing faces: Studies on the cognitive aspects of physiognomy. Journal of Personality and Social Psychology, 78, 5, 837-852.
Ekman, P., & Friesen, W. V. (1971). Constants across cultures in the face and emotion. Journal of Personality and Social Psychology, 17(2), 124–129.
Sergent, J., Ohta, S., MacDonald, B. (1992) Functional neuroanatomy of face and object processing. A positron emission tomography study. Brain 115, 1, 15–36.