Deleting the Little PrimalTap-plug didn't help. And the noise appears to come at random channels and buses, not only at the primaltap-bus, as I originally thought. Upped the buffer to 1024 and disabled all effects - the result of this is that audio plays, but there's still random static-like crackling on channels and buses, even though the CPU meters are barely showing any activity.

Here's a screenshot of two tracks that are consistently giving me trouble. Note that the top one only has Slate VCC on insert, while the bottom track doesn't have any insert effects. https://dyp.im/IveQLhtqXV

Inspected the audio clips of the tracks that were giving me trouble, and noticed that they (for unknown reasons) were reported as being 32 bits, while the rest of the project files were 24. Ran the "change audio format"-command set to 24 bits from the project menu - problem solved.