I'm pretty sure the only thing that would be affected under heavy load/overwrite activity would be performance. Not enough space for certain operations may slow down the operations, but I know of none that would make a LUN go offline as long as snapshots (volume based not snaps on the LUN) are not involved.
Overall for the initial question, a LUN or volume will never go offline just because the filesystem of the host is full as long as the LUN / volume on the NetApp is not full. This might cause the OS to get timeouts, errors or take it offline from the OS side, but on NetApp side if no other space in the volume is used (for example because of snaps) then a thick LUN thats 95% the size of a volume will never exceed its allotted space therefore the volume can never get "full"
Of course a full LUN or volume often corresponds to the host filesystem getting full, but you have to see them as two seperate issues.