A common origin for two languages is a concept that has been proposed and theoretically grounded within the comparative method invented at the beginning of the 19th cent. by Rask, Bopp and Grimm. Two languages are considered genetically related if and only if they present patterns of regular phonological correspondences in words with similar meanings. No other feature is required, or is enough, to prove the genetic relatedness. Typological similarity, even as striking as the one between Japanese and Korean, does not prove the genetic relationship. There are languages that are typologically similar but genetically unrelated. On the other hand, genetically related languages diverge very easily in their grammatical structure.
No one has been able so far to provide a convincing set of regular phonological correspondences between Korean and Japanese (less so for the alleged "Altaic family" in its entirety). Good attempts have been made in such respect, but eventually they have not met with general acceptance. It's a long standing controversy. It started with the Etymological Dictionary of Altaic Languages, a huge work by Starostin, Dybo and Mudrak (eminent Russian specialists in a big number of Altaic languages), then continued with a vehement, but meaningful, criticism by Vovin (a specialist in Japanese) and eventually concluded with a rebuttal by Dybo and Starostin Jr. The rebuttal is not convincing in many points, as has been pointed out by Ciancaglini, who makes also a good introduction to the Comparative method and briefly sums up the history of the Altaic controversy.
Generally speaking, the null hypothesis concerning the genetic relationship between languages is always negative. The burden of proof is on those who claim that a relationship exists. Therefore, answering your original question, we don't need special arguments for proving that Japanese and Korean are unrelated, but we need arguments for the opposite claim.