Quick A: Delete the r-value reference overload of the function.

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const-reference qualified member function

A temporary can be bound to a const& qualified object and the ref-qualifier effectively qualifies the implicitly passed object (*this). If you want to prevent calls on temporaries but allow lvalues, you can = delete the rvalue reference overload and implement the lvalue version. Using const qualified reference qualifiers for both operators requires just one implemented and one = deleted implementation:

class File {
    // ...
    FILE* _file;
public:
    operator FILE*() const&& = delete;
    operator FILE*() const& { return this->_file; }
    // ...
};

The net-effect is that you can use the conversion only for objects to which you go an lvalue:

int main() {
    File       f;
    File const cf{};

    FILE* fp = f;              // OK
    FILE* cfp = cf;            // OK
    FILE* tfp = File();        // ERROR: conversion is deleted
    FILE* mfp = std::move(cf); // ERROR: conversion is deleted 
}