Almost anything is ionizable, if you hit it hard enough...
In general, whether or not an organic group is considered ionizable depends on the conditions you are using. Some functional groups like carboxylic acids, phenols, and amines are fairly easy to ionize with mild acids or bases. Others like alcohols and thiols are easy enough to ionize that ionization plays a major part in their reaction chemistry. Amides, terminal acetylenes, and the positions alpha to keto carbonyls are ionizable enough to be useful in some chemistry.
A good table of pKa's of organic compounds is probably the best place to start when looking for ionizable functional groups.