In my theatre the Producer would have final say on some decisions especially those that would impact budget, theatre policies or operating procedures. For example the Directory might make a creative decision that he wants certain costumes for the show, but if this would push the show over budget then it's the Producer's responsibility to challenge that decision.

Fortunately in my group the Directors and Producers normally work well together so it shouldn't ever come down to the Producer having to absolutely forbid something, some sort of comprimise would be worked out.

It's my experience that many good theatres are run by benchmarking, by looking at "best practices" and then trying to fit things to our own situation, our own needs.  If you look at professional theatre of producing a Broadway play and then take what we can and make it our own, use some of the principles for our own needs.  In a Broadway show, the producer is the boss.  Without getting too grandiose with exagerated examples with multi-million dollar budgets and many producers and associate producers, etc.  But, for basic purposes to keep it simple: the producer is the general partner that raises all of the money.  The producer gets 50% and is called the general partner of the corporation.  All of the investors split the other 50% (depending on their investment) and they become the limited partners.  The producer is the boss and starts the process.  He or she buys the option of producing the play from the author(s) and the producer hires everyone: director, designers, stage manager, ad agency, negotiates deal with theatre, the producer is the overall boss.  Some times the producer fires the director and brings in someone else, but the producer has final say over everything.