I don't think so, it just reduces your standing with them.
I've been working hard at overthrowing the controlling faction in my system and have reduced it's influence significantly by doing missions for all the 'other' factions. In other words there is only 100% influence total in the system so by raising the others, the controlling factions has to drop.
Currently my system has reached civil war, but strangely the controlling faction is not yet involved.