What over-the-counter drugs will treat symptoms associated with cough variant asthma?

None. You need to see your doctor regarding this. Many people with cough-variant asthma actually needs a controller drug such as low-dose inhaled steroid or singulair (montelukast) to control the cough. Cough medicine is not recommended.
There isn't any. Unfortunately there isn't any medication specifically to treat cough-variant asthma that is available otc. You can, however, use some cough suppressant such as delsym/robitussin etc...But these only suppress the cough, maybe. So if you have prolonged, persistent cough, i suggest getting evaluated/treated. Reflux and allergies sometime cough symptoms. So, check with doc. Good luck.