I thought this thread might devolve into bad jokes and quotes from obscure movies, but instead it’s been very educational.
“Supplies!”
This is not the same thing. First of all, j or /dZ/ does exist among many Spanish speakers in the United States, and it is spelled with the letter Y. This would be akin to a German always pronouncing W in English as it would be in German or English “th” as /s/ or /z/ (“zis sing” for “this thing”). Or an English speaker always pronouncing J as in English when speaking German. This is easily chalked up as a mistake or a foreign accent or an imperfect knowledge of another language.
The existing only in musichall comedy “accents” I’m talking about is an exact, one-to-one, consistent exchange of W for V and V for W. I simply do not believe you can produce a German who would speak English like that. Or an East Asian who exactly, consistently exchanges R for L and L for R. So unlikely as to be unbelievable.
A few years ago I was driven to work on a shuttlebus by an elderly asian man, who announced on the intercom “Next stop, Bowees”, instead of Ballys.
My understanding with regard to Korean is that the Riul character is usually pronounced as L if it is the very first consonant in a word, or if it is at the end of a syllable that is followed by another consonant, and pronounced as R in all other cases, with occasional exceptions for loanwords.
Do fight my ignorance 
Yes, we met in another thread. Professional linguists, including Koreans, recognize those particular sounds in Korean as I have described them in that thread.
So, are the /L/ and /R/ sounds written the same way in Korean?
They are represented by the same letter, yes - the letter Riul ( ㄹ ). It’s usually pronounced close to an “r”, but in some situations it becomes something more like an “l” (basically when two Riul meet without any other letters in-between). See Monty’s examples.
As for the last name “Ii” being spelled Lee, Yi, and Rhee, I’ve often wondered that myself. The Chinese character is pronounced both Rhee and Ii (ee), depending on where it appears in a word. At the beginning of a word it becomes Ii, but if it comes after the first syllable it becomes Rhee. In Korean family names come first, so it is always pronounced as Ii, but I’ve often wondered if some people, upon coming to America, changed it to Rhee since in American people put their family names last. (This is a completely personal obversation and I have nothing to back it up.)
I have noticed that different Lee clans will romanize their last names differently as well. I don’t know if that stays consistent, though.
Gah. “Ii” is two iis, not an L and an i. Curse this stupid font.
So is it that, rather than being two separate phonemes in Korean, they are both allophones of the same underlying phoneme? Paladud’s explanation would seem to indicate that it is. In that case, we could still expect some confusion and mistakes from native Korean speakers regarding when to use /r/ and when to use /l/ in English.
Incidentally, looking at Wikipedia (and Wiktionary), it seems the Korean ‘r’-type sound would be more of an alveolar flap (as in Spanish as remarked upon by Bear_Nenno) than an alveolar approximant as in English. I understand that Monty is well-versed in matters both linguistic and Korean, so I’m inclined to bow to his knowledge, but it does seem as though there may have been something to what audiobottle was saying.
I am not a linguist, but yes, there is a lot of confusion (for example, my students used to pronounce “really” as “lee-uh-lee”). And the “r” sound is a bit different from the English “r” sound in acutal pronunciation (although I make no claim as to what the technical linguistic differences are). There is a bit more of a roll to it. I am certain that there are two distinctly different ways to pronounce Riul, as Monty said in his previous post, but I’m also pretty certain that there is some kind of difference between the English “r” and the Korean approximate of it.
Also, Koreans don’t have the F or Z sound. Had a hell of a time teaching my wife to properly make the F sound. They usually substitute the P sound when saying english words that need an F.
I’m not a linguist, so I dont really know the difference between a phoneme and an allophone.
But that symbol is similar to an American “c”. Not in sound, but in the fact that it can make two seperate sounds (‘s’ sound or ‘k’ sound).
Knowing which sound to use comes from knowing the language, a few little rules, and learning the exceptions from experience with the language. So Koreans will know when it’s an L or when it’s an R in Korean. But, when there is an English word spelled with Korean letters, they will not know if it’s an R or an L. That’s the root of most of the confusion.
Generally, Rs come at the begining of the syllable, and Ls come at the end. So English words which do not follow this rule will also cause confusion or at least make the word less natural for a Korean speaker to pronounce.
That’s my understanding anyway. I’m self-learning Korean, so I’ve had no formal training. Just books, programs and practice.