Pure fats are just that: Pure. Fat. Fat does not support microbial growth, so there’s no reason to chill your lard, canola oil, olive oil, or any other cooking fat, for that matter. Plus, it can pick up flavors and odors from other stuff in the fridge if it’s not stored in an airtight container.
The only exception is fat you’ve rendered yourself, left unstrained. Take bacon grease: Strained, it will last as long as any other fat, but I rarely strain mine, because I like the little meaty bits. But those bits are made of protein, and bacteria loves protein. For this reason, I keep my unstrained bacon grease in the fridge (and you should too).
Rancidity is the other thing to worry about with fat, but keeping your cooking oils out of direct light and at cool (not cold) temps can stave it off. When in doubt, give the fat a sniff. If it smells soapy, metallic, or bitter—or there’s an unpleasant sticky reside on the container—it’s probably best to toss it.
