I'm pretty sure the Union controls the rail lines north of Chattanooga that would feed it, so there isn't any reason why Buell couldn't hold the city. There is always the possibility he will lose a battle, so it's not certain, but very plausible he can hold it.
I see two major things that would follow Buell's capture of Chattanooga. First, it allows the Union to seize the rest of Tennessee - the Unionist hotbed of eastern Tennessee. An expedition will be sent to Knoxville as soon as possible to open up that area to Union recruitment. If the Union can hold all of Tennessee, then Lincoln may move to return TN to civil government under Andrew Johnson ASAP.
The second is that the Union threatens to move on Atlanta. Might any forces be diverted from the Army of Northern Virginia to defend the city? If so, it might affect the critical battles in late 1862 - 2nd Bull Run, Antietam, and Fredericksburg. It sets the stage where the Union might have a very strong 1863 - the potential to take Vicksburg, Atlanta, and put more pressure on Virginia would severely hurt Confederate morale if they happen.
Buell taking Chattanooga might shave off a good six months or more off the war.