If you have the time and aren't in a hurry then do it. You need a minimum of 5 days but better to have 7 to travel the entire length if you can get out and enjoy any of the views. I've traveled every section multiple times and always find something else to see. I've camped on the southern end from Ashville to the southern end for over 40 years and it's still one of my favorite places to visit within 500 miles of Charleston.
Have done two long sections, Ashville to Waynesboro and just last Oct Front Royal to Roanoke. However when I retire and can take my time, will do the entire length from Front Royal to southern end in a single trip over a one week period.
As mentioned, the tunnels are arched and have the minimum height measured at the white line. By the time you move over 3' to hug the white line you are another 1' - 2' taller and by the time you get to the ac in middle of your rig you will have another 1'. While there is a tunnel or two where you will want to cross the yellow line to take the center of the tunnel, you do not need to, just hug it.
The fuel is real easy as you can get fuel near each interstate crossing or big towns. The longest dry section is from Ashville to southern end, but even then you can get fuel by dropping off at 19/23 and going a few miles towards Waynesville.
In regards to campgrounds, they are nice just be careful as they will fill up on Friday, some on Thursday. So I either make reservation for the weekend or if at a First Come, stop early Friday and stay until Sunday.
Make sure you check out http://www.blueridgeparkway.org/ as they have a lot more detailed information. Particular attention to the hiking trails as they list all the trail starting points, some of which are not mentioned even at the overlook. There are a number of short easy 1/4 mile hike to get the views.
Mabry Mill is great for a morning breakfast. Works well if you spend the night at Rocky Knob, get up early and head south a short way to the eastern facing overlook to watch the sunrise, then a few more miles for breakfast.
Other place to eat is Pisgah Inn which usually has a favorite good chief and an unbelievable view. It is a nice short walk from the campground to the restaurant so if you can walk about 1/4 mile each way you wouldn't even have to move the rig.
I could go on, but think you can tell I get excited about the BRP. Happy trails and enjoy.
2015 Ram 3500 4x4 Crew Cab SRW 6.4 Hemi LB 3.73 (12.4 hand calc avg mpg after 92,000 miles with camper)
2004 Lance 815 (prev: 2004 FW 35'; 1994 TT 30'; Tents)