I am considering using virtual inheritance in a real-time application. Does using virtual inheritance have a performance impact similar to that of calling a virtual function
Are you sure you mean virtual inheritance? If so, it's identical to the cost of a normal virtual function call. The vtable chained search just follows a specified path.
You said that this was at startup. Your disk overhead (from simply loading your code into memory) is likely to require orders of magnitude more time than the half-dozen instructions or so for vtable lookups. I'd be somewhat surprised if you could profile this and detect a difference.