It's true that "positive" is an adjective and shouldn't be used to modify a verb. I definitely wouldn't recommend using "think positive" in a formal setting. "Think Positive" isn't really grammatically correct, but it's also very common, arguably more common than "think positively." This is just one of those cases where actual usage of language differs from formal grammar.

Both forms are possible. There are many situations, particularly in set phrases involving imperatives, where it's acceptable to use what's known as a 'flat adverb'. Common examples of imperative collocations using flat adverbs are 'Sleep tight' , 'Hold tight', 'Say it loud and clear' and 'Take it easy'. I would put 'Think positive' into this category. In fact, there are some instances where a flat adverb is the only option, for example, 'do something right' or 'do something wrong'. The 'ly' adverbs 'rightly' and 'wrongly' exist, but they have different meanings. Don't assume that an adjective form modifying a verb is necessarily poor, lazy, dialectal or informal English. In some cases it is - as in 'You done that real nice', for example - but not in every case. 'Think positive' is an instance where a flat adverb is fine.

I think that 'think positively' sounds more natural. Not sure about 'think positive', on its own it sounds a little bit odd. I'd probably say 'be positive' instead.

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.

Positively should be used as an adverb to modify the verb "think". The "positive" in "Think positive" is really acting as an adjective, and is more like saying "You should think positive thoughts", just shortened down to "Think positive". It's not really correct English, but as you state, it is used commonly and you will hear it in common conversation.