during an aram match,, i occasionally used my W 
on a jarven with banshee and for some odd reason it would not get rid of his banshee when i used it (even with rylias). im not sure if its only jarven or if it affects other champions with banshees.
Some champion abilities don't pop Banshee's Veil, but the effect of the ability will be reduced(?).
All champion abilities that does constant AoE damage in an area will not pop it. Anivia's R and Cassiopeia's W for example.
Like the other poster said, ground targeted DoT abilites (like Morgana W, Annivia R, Miss Fortune E) don't ever pop banshee's veil (or Sivir/Noctrune spellshield). (they don't reduce the effect of the ability in any way either). You can buy 
if you need to pop Banshee's, as the passive on Luden's will pop veils.
Some AOE DoT abilties do pop banshee's veil though (MF ult, Lucian ult), in that case it will stop one "wave" of damage but subsequent waves will hit.
