Alright, so my familiarity with this area primarily comes from Vanessa LoBue's work. And what I get from her research is that we don't really know if certain fears are innate or acquired.
LoBue seems to favor a prepared learning model, which is just as it sounds. Infants are not born fearful of things like spiders, snakes, and heights (Adolph, Kretch, & LoBue, 2014; LoBue et al., 2012), but they are biased (prepared) to learn to fear them. She has supported this theory by showing that children are more rapid at attending to snakes than other stimuli (e.g., flowers, frogs, caterpillars), independent of snake knowledge/exposure (LoBue & DeLoache, 2008). Infants are also biased to associate undulating snakes, but not other stimuli, with negatively valenced sounds (DeLoache & LoBue, 2009). However, this wasn't true for static images of snakes.
Importantly, these stimuli are (ostensibly) evolutionarily significant (snakes, spiders, but not lions, tigers). However, the evidence for their threatening presence in human evolution, at least for spiders, is lacking (LoBue & Rakison, 2013). This has been a major criticism of the prepared learning model: which stimuli are 'prepared'?
LoBue suggests that the bias toward snakes might be mediated by low-level perceptual characteristics that pop out (i.e., curvilinear features) (LoBue, 2014). However, it's not clear if this visual saliency leads to actual fear learning of snakes (LoBue & Rakison, 2013). Moreover, this suggests that any predominantly curvilinear stimulus might bias attention.
Aside from her work, there are loads of mixed findings in this area that don't point to a clear answer. For example, there are studies that report a general attention bias for animals in general, threatening or non-threatening (e.g., Lipp, 2006). Plus there are several outstanding methodological problems in much of the past research.
Frankly, there just needs to be a lot more research in order to figure this out. Although I would argue that the "non-associative model" (that infants don't need to learn, in some way, to fear snakes, spiders, etc.) is out of fashion and not well supported. But for the most recent reviews, see LoBue & Rakison (2013) and LoBue (2013).
Adolph, K. E., Kretch, K. S., & LoBue, V. (2014). Fear of heights in infants?. Current directions in psychological science, 23(1), 60-66. doi: 10.1177/0963721413498895 PMCID: PMC4175923 PDF: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4175923/pdf/nihms574402.pdf
DeLoache, J. S., & LoBue, V. (2009). The narrow fellow in the grass: Human infants associate snakes and fear. Developmental science, 12(1), 201-207. doi: 10.1111/j.1467-7687.2008.00753.x
Lipp, O. V. (2006). Of snakes and flowers: Does preferential detection of pictures of fear-relevant animals in visual search reflect on fear-relevance? Emotion, 6(2), 296-308. doi: 10.1037/1528-3522.214.171.1246
LoBue, V., & DeLoache, J. S. (2008). Detecting the snake in the grass: Attention to fear-relevant stimuli by adults and young children. Psychological science, 19(3), 284-289. doi: 10.1111/j.1467-9280.2008.02081.x
LoBue, V. (2013). What are we so afraid of? How early attention shapes our most common fears. Child Development Perspectives, 7(1), 38-42. doi: 10.1111/cdep.12012
LoBue, V., Bloom Pickard, M., Sherman, K., Axford, C., & DeLoache, J. S. (2013). Young children's interest in live animals. British Journal of Developmental Psychology, 31(1), 57-69. doi: 10.1111/j.2044-835X.2012.02078.x
LoBue, V., & Rakison, D. H. (2013). What we fear most: A developmental advantage for threat-relevant stimuli. Developmental Review, 33(4), 285-303. doi: 10.1016/j.dr.2013.07.005
LoBue, V. (2014). Deconstructing the snake: The relative roles of perception, cognition, and emotion on threat detection. Emotion, 14(4), 701. doi: 10.1037/a0035898