Local refining is setup for refining heavier crudes. They can process sweet domestic crudes just fine at a technical level. It would be an economic loss due to underutilized stuff like Cokers, and likely at somewhat reduced overall throughput. Light crude is typically more expensive than heavy which accounts for much of the theoretical economic loss, but perhaps that will be inverted for some time if trends continue.

You would lose some of the bottom of the barrel products like asphalt and the high sulphur products sour crudes have as well, but I'm unsure of how impactful that would be in practice.

I'm certainly no petroleum engineer so I'm sure someone will be along to correct me - but I looked into this when I kept seeing this trotted out. You can definitely refine domestic light crude oils in local refineries setup for heavier crudes. The resulting products will simply be more expensive due to the refinery operating less efficiently. Self-sufficiency for fuel products at least is likely not a major concern for the US if the shit hits the fan for real.