Does polenta 'go off'? I have some polenta in the cupboard that I can only assume is a family heirloom.  The best before date has long since perished but the contents look okay.  

Does polenta ever stop tasting of polenta?
Yes, it can go bad.

It's pretty much the same as the way flour can go bad: the fats can go rancid. Grains are mostly starch, but they contain enough fat for rancidity to be a noticeable problem. There's a huge amount of variability in how long that takes, though. If it's airtight and kept in a cool, dark place, it'll last much longer than the best before date. But if it's been exposed to air, light, or heat, it can go pretty fast, perhaps even before the best before date.

So, smell it. If you detect any unpleasant smells, you probably don't want to eat it. Even if you don't smell anything obvious, it could still be subtly rancid, so if you're planning on serving it to anyone, you might want to cook a small bit to test.