If either Tennessee or North Carolina had not seceded (notice I'm refraining from the saying "voted against" since only Texas really had a popular vote to determine whether the state would or would not secede) then it would have reinforced the rather unrealistic view held by Lincoln and other Republicans that the majority of Southerners did not support secession and would somehow work to bring their states back into the Union before a major conflict would arise.
South Carolina, though was dead set on secession by late November of 1860 so the Bombardment of Fort Sumter would have occurred nonetheless. Lincoln may have delayed somewhat his call for 75,000 volunteers under the belief that the secession movements lacked wide spread popular support. But eventually he would be forced to act, because although secession came about due directly to the sectional dispute over slavery, Lincoln's immediate aim was to suppress the Rebellion and preserve the Union. More than likely (assuming as per the OP that it is NC that refuses to secede) Lincoln will offer to send federal troops to North Carolina in order to maintain order and keep the state in the Union. The Gov. of NC will refuse and instead declare state neutrality as Kentucky did and Missouri tried to do.
A neutral NC helps the Union while really hurting the Confederacy. Virginia is cut off and the Confederate capital will stay in Montgomery. Even more important for the Confederacy, North Carolina is an important rail link between the Deep South and the Upper South (as is Tennessee). A mini-Civil War will occur in NC between pro-secessionists unwilling to accept neutrality and pro-Unionist who loath the idea of secession. More than likely Lincoln will be smart enough not to send federal forces but the bifurcated Confederacy will not have the luxery of allowing NC to remain aloof. The Gov. of SC will send troops to aide the pro-secessionist faction, and Lincoln will respond by sending troops to assist the pro-Unionists.
So essentially it will be Kentucky and Missouri rolled into one lovely ball of confused fighting, back-stabbing politics and brutal guerrilla warfare. Of course the Confederate states will look like a bunch of even bigger hypocrites than they already do. At least if its Tennessee that refuses to secede than they can form a local alliance with Kentucky which may actually save their states from being battlegrounds.
Benjamin