I have the same problem!
If the problem persist, that means your sdcard is full. Try move some files on your sdcard to extra sdcard
Beta1 was the first LP I installed on top of a long series of perfectly working OmniROM KKs. Most things worked as advertised, but APP2SD didn't work, even if there was plenty of GB left on both the internal and the external sdcards ("no storage available", even if 20 out of 32GB were free).
Strangely it showed me my 32GB external SDCARD under SDcard in settings/storage only, instead of the ca. 10GB of internal simulated SDcard on eMMC: Effectively it limited app storage to the 2GB /data partition on eMMC, while multi-media content from the external SDcard remained available (removing the external SDcard simply meant no SDcard was visible for the app manager).
At the same time Solid Explorer had no issue at all distinguishing between the internal and the external SDcard or indeed /data and I was freely able access, add, move or delete files between them.
I looked through 'mount' and 'df' outputs, at the fstab file etc. and apart from a 'old' prefix used to refer to the internal sdcard I couldn't find anything wrong.
Now since the majority of the users of this ROM don't seem to have any issues, but at least two do, the problem is more likely dependent on the state of the device, which is why I simply went ahead and tried to a) see what removing the external SDcard would do (no improvement!) b) see what wiping the internal SDcard might change.
Unfortunately wiping the internal SDcard didn't change anything for the better, actually after I decided to see if I could use ext4 instead of VFAT (I'm hoping to get TRIM support for speed), LP Beta1 would no longer enter 1st time setup (after the boot animation finished, the screen would simply stay black).
Of course I tried reformatting with VFAT, "default" (which I assume is VFAT) and plenty of wipes and factory resets, but that didn't change a thing.
I went back to Omnirom, which was easily installed via CWM from the external SDcard and there all storage worked perfectly, including the internal SDcard using ext4 (but no TRIM support in the kernel

)
Of course I tried immediately to go to LPb1 again, but for some strange reason, LP won't ever come alive after the long inital boot (had it lie for longer than an hour).
So I guess I'll have to wait for Beta2 or an OmniROM LP or I'd have to try going back to some "factory image" via ODIN first (can't say I enjoy that very much).
I've moved my Nexus 10, Nexus 7, Asus TF101, and a son's Nexus 4 from KK to LP and am quite impressed with LP's performance even on older and RAM limited devices, especially on devices with fast eMMC like the TF101 and the Nexus 10.
Timduru's rendition of LP on the TF101 has that old warhorse really kick ass and since the Galaxy Note has a bit of extra hardware OOMPH in any direction, I'm pretty convinced LP on the N-7000 should be not bad at all!