It is correct English to use an adverb to modify a verb. In a business letter, a magazine article, a committee report, a speech to an audience, and--of course--in an English class you should use the correct form.
"Think positive," however, has become a stock phrase in itself. The phrase is "think positive," not "think positively."
In practice, in real life, it is very common for native speakers to drop the "-ly." People do this for various reasons. They may not use much formal or written English. The language may be evolving, and there is definitely a trend to shorten words and phrases. Or, the use of strictly correct English may be seen as a mark of high social class and people may want to distance themselves from the elite and present themselves as "regular guys."
You will often hear phrases like "she answered right" or "he won big." A hit song of the 1950s was "Love Me Tender" and people complained about the "bad grammar."
It's just one of many cases where the careful language of well-educated people who work with words and write a lot, is a little different from the everyday language of adequately-educated people who speak everyday talk.