You would be required to drop values that aren't moved elsewhere. Currently Rust has a one-bit reference count which dynamically tracks which values need to be dropped. Linear types would transform this into a static property that the compiler checks.

Part of the pain would come from conditionals:

    if something { 
        func(val1);
    } else {
        func(val2);
    }

This would be disallowed because the liveness of val1 and val2 can't be statically known after the conditional.

I guess it is only for ones with nontrivial destructors.

That makes the "trivialness" of a destructor part of the interface, which is a price. But then from my experience in C++, we pay that price anyway, because to have any sort of assurance about what you are doing, you need some clue about what the destructor does.