We investigate social transmission of behavioral innovations between predators in two classic predator-prey models. We assume that innovations increase predator attack rates or conversion efficiencies, or that innovations reduce predator mortality or prey handling time. We find that a common outcome of innovations is the destabilization of the system. Destabilizing effects include increasing oscillations or limit cycles. Particularly, in systems where prey are self-limiting and predators have a Type II functional response, destabilization occurs due to overexploitation of the prey. Whenever instability increases the risk of extinction, innovations that benefit individual predators may not have positive long-term effects on predator populations. An additional consequence of instability is the maintenance of behavioral variability among predators. Interestingly, when predator populations are low despite coexisting with prey populations near their carrying capacity, innovations that could help predators better exploit their prey are least likely to spread. Precisely how unlikely this is depends on whether or not naïve individuals need to observe an informed individual interact with prey to learn the innovation. Our results offer perspective on the potential role of innovation in biological invasions, urban colonization, and the maintenance of behavioral polymorphisms.