I wouldn't worry about "running out" of room in our brain. We often forget things just because we don't need to know them, similar to . There are plenty of human savants that display apparently "unlimited" capacities of certain forms of memory, such as Hyperthymesia where one has a seemingly perfect amount of recall of autobiographical details (not to be confused with the largely debunked notion of "photographic memory").
It appears we forget things not because we're low on space, but because we don't need that information (not that there's intent, it's just practical). Information often accessed is strengthened, information never used is weakened and lost. Similarly we physically prune neurons and connections we no longer use in Synaptic Pruning.
Back to the issue of survivability; it's possible that operant conditioning a wild animal could cause them to incorrectly react to natural stimuli. The biggest risk here would be Generalization of stimuli, like Little Albert becoming (temporarily) afraid of all things white and fluffy instead of just the rat he was conditioned with. If you trained a dog that brown furry things means I should attack and he generalizes this to a bear, it won't end well for the dog of course.
This would generally be less of an issue than you might fear however; an animal will quickly forget the conditioned response due to Extinction provided the Conditioned Stimulus is no longer paired with the Unconditioned Stimulus. Repeated exposure of the CS with no US will eventually extinct the reaction; reinforcement learning goes both ways.
It's even more unlikely that completely artificial conditioning will result in prolonged harm to survivability. This is because if the Conditioned Stimulus is never presented, the conditioned response will simply not be activated. The CS won't even need to be extincted, the animal would likely just forget eventually.
Now the one thing I would worry about is Extinction of natural defense/predatory responses. If an animal has learned through experience to do or avoid doing something and, through domestication/etc they no longer feel the urge to run from predators it will of course impact their survival.
It's impossible to say one way or another really, but I hope that helps you understand what's potentially at work here.