Now that I know how to call people a tiger, I think I've learned all the Italian I'll ever need.
The word is feminine but refers to both male and female animals. My guess is "una tigressa" would refer only to the female animal.
Don't confused gender and sex. Languages with gender don't have a logic about gender. There is no reason why tigra is feminine in Italian and masculine in French.
Can I call my boyfriend a tiger ;) or is it not appropriate because of the gender?
I think you can. The gender of the noun isn't related to the gender of the subject.
It just is. Words in Italian are either feminine or masculine - and there's not a lot of rhyme or reason why one word is feminine and another word is masculine.
The Italian words for most animals have masculine gender (un cane, un cavallo, un topo), but not all (una tigre, una scimmia, etc)