As a Japanese-American I've put a lot of thought and reading into this. The short answer is, as others have written, it is not proven either way -- however the evidence is mounting recently in favor of a relationship. There was a time when a relationship between Korean and Japanese was thought to be likely, then linguists moved away from the hypothesis and it was considered mostly debunked, but more recently it's gained traction again with the advent of new computational linguistics evidence. One of the most prominent early proponents of a linkage is the linguist Alexander Vovin, but he later rejected the hypothesis and now believes the languages have independent origins. However, I'd say one of the most impressive new researchers in the area, Martine Robbeets of the Max Planck Institute, and her research group, have done remarkable work tying together genetic, archeological, linguistic, and computational models with compelling evidence of a genetic link between the languages.
Note that it is important when reading papers on this topic to note the dates -- every researcher in this area has steadily evolved their views on it, and the most recent papers will be the most reliable indicators of their current opinions.
Here is an interview with Robbeets where she outlines some of her groups' research on Japanese and related languages:
What I find particularly impressive about their research is they were able to use computational models to reconstruct past versions of languages and using that approach they conclude that Japonic and Koreanic had a shared ancestor, but more importantly they also were able to make a prediction that the original homeland of the shared ancestor of not only Japonic and Koreanic, but Mongolic and a number of other language groups would have been millet farmers living in a specific valley in Inner Mongolia 10,000 years BP -- and sure enough, archeological evidence shows millet farming origination in precisely that valley at precisely that point in time. An extremely impressive cross-disciplinary corroboration of their results.
Despite these varying views, however, it's well-accepted that proto-Japonic speakers once lived on the Korean peninsula -- this is accepted by both Vovin and Robbeets and there is considerable evidence for it. Japonic toponyms have been found in phonetic accounts of words used on the peninsula during the Three Kingdoms period. There is some discussion of this here:
but there are many other references.
Furthermore, another computational linguistics analysis of Japonic dialects (including Ryukyuan) concludes that they all diverged from a common ancestor approximately 2,200 years ago, which corresponds approximately to the archeological record of the largest number of bulk of the Yayoi migration to the island of Kyushu. Here's a news story covering this study, from there you can find the original paper:
This pretty much excludes the hypothesis the Japanese language evolved on the islands -- it was clearly imported by the Yayoi, and the other evidence ties this in -- proto-Japonic speakers once lived in Korea and came from there.
However, there is the problem that the core vocabulary of Korean and Japanese do not sound similar, including base words such as counting words and so on. There are very few obvious cognates except for words known to have been imported from China. For this reason most linguists still believe the languages are related only via sprachbund. I believe, however, that Robbeets' team is correct and this conventional view is not correct. Robbeets has done remarkable work showing strong evidence of their genetic relationship, but I would like to add another argument in favor of her conclusions.
I would posit two things -- the subject - object - verb word order plus the use of particles, adding in the very unusual use of wa/ga in Japanese or eun-neun/ga-i in Korean, is an extremely rare grammar in the world. There are many languages with subject-object-verb word order but almost no languages with all of these features put together. For this reason it is exceptionally unlikely that if the two languages converged by sprachbund that the original versions of either languages had a similar grammar. There just aren't many languages with such a grammar anywhere on the planet. In particular, even very subtle features such as the use of the topic and subject markers I noted above are pretty much identical in the two languages, yet this is also an incredibly unusual grammatical construct and it is furthermore extremely difficult to explain even for a native speaker. The fact that this grammatical construct exists in both languages and is pretty much identically used, down to subtle details, means the sprachbund, if it occurred, was so deep that it managed to transfer not only the overall grammatical structure but some of the most tricky and idiosyncratic details of usage.
Because this grammar is so unusual, I would posit that if they converged due to sprachbund, whatever the original grammars of these languages were, they would almost certainly have been quite different to begin with.
This in itself does not exclude the possibility of sprachbund -- however I think there is one other fact which ironically has been used to show the languages must not be genetically related, and that is the fact that there are very few if any clear cognates between the core vocabularies of either language. Here's the fundamental problem: if these two protolanguages, proto-Japonic and proto-Koreanic, started out with different grammar and converged through close interaction over a long enough period that they came to have nearly identical grammar -- why AREN'T there more cognates? With such a presumably close relationship, one would except a very large number of borrowed words in one or the other direction. Yet there's very little evidence of this at all.
One thing that is well-known in linguistics is that vocabulary changes more quickly than grammar. Grammar tends to be stable; vocabulary is fungible. Yet here we are expected to believe that these two languages converged to an extremely unusual, nearly unique grammar, without having also borrowed nearly any vocabulary at all in either direction? To my mind, this on its face makes the sprachbund hypothesis exceptionally unlikely.
The archeological evidence also doesn't really support this hypothesis. It's now known that the majority of Yayoi migration to Kyushu occurred sometime between 800 BCE and 300 BCE (it was once thought the Yayoi didn't arrive until 300 BCE but there are now known archeological records of earlier arrivals hundreds of years before). We also now believe this corresponds to the time when proto-Japonic was likely most spoken on the Korean peninsula. However, archeological evidence also shows proto-Koreanic speakers arrived on the Korean peninsula around 300 BCE. It is very likely the reason for the Japonic toponyms is the two groups overlapped and it is also very likely many ancestors of modern Koreans are descended from these proto-Japonic speakers. But the time period of overlap between proto-Japonic and proto-Koreanic was quite recent -- sometime from 300BCE to perhaps the early Three Kingdoms period. Note that an alternative hypothesis that the sprachbund may have occurred when proto-Japonic speakers lived further north, perhaps in an earlier time, seems to be excluded by the fact that the Japonic toponyms only seem to be attested in the southern part of the peninsula.
If the proposed sprachbund occurred during this overlap it's hard to explain the fact that there is little evidence of shared vocabulary from that sprachbund which is relatively recent in linguistics terms. Also, many Yayoi had already migrated to Japan before the Koreanic speakers even arrived in Korea -- yet there's no trace of whatever language they may have spoken -- the computational evidence is modern Japanese diverged from a single ancestor around the time of the Yayoi migration, with a founder effect. If Japonic had somehow evolved during this period of overlap and changed its grammar radically, it's hard to understand why there's no evidence of this and in fact it appears the modern Japonic language family came from a bottleneck event and a single ancestor founder effect at that time.
All of this evidence is far more consistent with the Robbeets proposal. In her team's account, Koreanic and Japonic speakers were once one group, but they split apart for a couple thousand years give or take, and re-encountered each other again in large numbers later, during the overlap period I mention above. During the period of separation the vocabularies diverged but the grammar remained consistent. Furthermore, Robbeets and her collaborators believe the proto-Japonic speakers encountered Austronesian speakers while living on the Liaoning peninsula, where they learned rice agriculture and may have imported new vocabulary. Since vocabulary tends to change much more rapidly than grammar, this would be consistent with what we observe: divergent vocabulary but shared grammar.
A recent (May 2020) whole-genome genetic study concludes that Koreans and Japanese branched off the same genetic root and are by far the most closely-related genetically to each other than to any other ethnic groups in East Asia:
The key graph from this paper showing their conclusions:
This is further evidence of the close genetic relationship between the two groups, which is also strongly consistent with Robbeets' hypothesis.
Finally, I'm going to point to Alexander Takenobu-Francis-Ratte's thesis in which he outlines a proposed set of patterns to explain how the vocabulary of Japonic and Koreanic may have diverged in a consistent way over the centuries and he reconstructs a core set of cognates from this of around 300 words. Vovin has strongly objected to his paper but I nevertheless leave it here as additional interesting work to add to this topic:
He is currently a linguist teaching at Furman University.