It is possible to get tonsillitis from kissing if the tonsillitis is viral or bacterial in nature. If the tonsillitis results from chronic conditions like allergies or sinusitis, it is generally not contagious.
The most frequent cause for tonsillitis is the repeated incidence of chronic infections. These cause the tonsils to swell and become uncomfortable. When the infection comes from a virus or bacterium, the onset is also accompanied by a fever, which means that an elevated temperature is a good test as to whether the case of tonsillitis is actually contagious. Even if the particular case is not contagious, the discomfort makes kissing much less enjoyable.