Deks makes many good points.
On a planet by planet basis, yes there would be some effect. But the Federation is massive, pulling in resources from 150 member worlds, who knows how many colonies, as well as allies that aren't even part of the Federation but simply maintain trading agreements.
And realistically the Federation wouldn't put all their eggs in one basket. They would be set up to the point that even if the Dominion had taken Earth and Mars, and destroyed Utopia Planitia, the war would have kept going.
As far as conscription and conquering a planet, by the time the Federation was in a position to start taking Dominion held worlds, the Klingon and the Romulan empires were also fighting with them. So while most of the fighting seemed to be taking place in the Alpha quadrant, the Federation wasn't shouldering the burden of the entire war and taking a planet wouldn't have been their sole responsibility. Raising a force to take a planet wouldn't have been a problem for three quadrant spanning powers that likely each have populations ranging in the trillions.
I seriously doubt Starfleet had to operate any sort of conscription service. I could see them offering some sort of basic training incentive program though. Something along the lines of singing up for a standard tour of duty as an engineer, medic, or security crewmen (some type of position that doesn't require the full Academy run) and when you're tour is up, you get a guaranteed placement at the Academy, forgoing the usual entrance exam.