I found eight examples of "chill the weekend out"(with quotes) on google, four of which are from this thread, two seem to be from an indian website, and 1 from myspace.cn (China based) websiteGrammatically you are changing an intransitive verb ("to chill out" doesn't take an object; you chill out yourself, you don't chill something else out) into a transitive one, and the object is "the weekend" rather than youself. This is perhaps illogical, strictly speaking, but nevertheless this sort of reversal does occur in casual and idiomatic speech, and in this case I think it works quite well. To me it's quite understandable and sounds natural. It's rather like using "chill out the weekend" in the same way we use "spend the weekend", but with an adverbial sense.
I notice there are a few (although only a few) examples of this usage on the web, so it obviously seems natural to at least some other people as well.