Should 'tiger' be capitalized?

No, the word 'tiger' should not be capitalised as it is not a proper noun (as shown in the title to this short post). A proper noun is a noun that serves as the name for a SPECIFIC place, person, or thing. 

An example would be the word 'Bengal' which is an historic region in the Indian subcontinent at the apex of the Bay of Bengal.

Interestingly, the most common species of tiger is written 'Bengal tiger', with the first half capitalized as it is a distinct place in Asia and the second half not capitalised as the word 'tiger' refers generically to the species of wild cat called a tiger.

You often see this 'hybrid' labelling of the wild cat species. For example, the Andean mountain cat or the Chinese desert cat or the now extinct Caspian tiger.

To distinguish them from common nouns, proper nouns are always capitalised in English. Common nouns are words for types of things, people, and places, such as 'dog', 'professor', 'city', and 'tiger'.

Finally, there is a certain amount of flexibility in this and English grammar has evolved and continues to do so. 

For example, Dr Desmond Morris in his book Cat World would write 'Caspian Tiger' or 'Bengal Tiger' with a capital 'T'. That's because he was capitalising the entire name which was normal in 1996 when the book was published. 

It sort of looks better if we are honest but it is grammatically incorrect nowadays.