I was interested in this question because I have always been a little confused about allergic responses, so I did some rather superficial research (i.e. I looked at some Wikipedia pages). As far as I can see there is no reason to think that Montelukast will have any effect upon IgE levels.
Montelukast is a leukotriene receptor antagonist. It is used to treat allergies and asthma.
Leukotrienes are lipid signalling molecules that are used by cells to regulate immune responses. One role of leukotrienes is to promote contractions in the smooth muscles lining the bronchioles. If they are over-produced they can cause inflammation in the airways leading to symptoms of asthma. Since leukotrienes act through a receptor, an antagonist of that receptor such as montelukast can reduce their effect.
IgE is involved in allergic responses because it interacts with Fc receptors on the surface of certain cells such as mast cells. If the IgE molecules are associated with one another through Ig-antigen interactions this leads to clustering of the Fc receptors which triggers an intracellular signalling pathway leading to mast cell degranulation - the release of histamine by fusion of storage vesicles with the cell membrane (i.e. exocytosis). Histamine causes constriction of the bronchioles via effects on smooth muscle, just like leukotrienes.