|Left: the letter Ö, on a boundary stone at the border between Austria and Germany, such as found in the surname “Göethe”, wherein languages that do not have the letter as part of the regular alphabet (such as English), the O-umlaut, the double dots above the letter (ö), is frequently replaced with the digraph "oe", thus rendering the name as "Goeethe" (English). Right: German Tomasz Kurianowicz discussing Goethe, to hear example pronunciation.|
Göethe → Goethe
“Goethe, whose name in charity to young pronouncers should be spelled Getty, was the father of affinity. If there is a more dangerous combination than a woman, a kerosene can and damp kindlings, it is a poet, an inquiring mind and the science called Chemistry. While tinkering with the science in an amateur way. Getty, the German Shakespeare, discovered, what every chemist knew, that there are certain atoms that possess elective affinities for certain other atoms; that is, they were intended by Nature to meet with and adhere to each other, forming a perfect whole.”In 2006, Irish-born, English-educated, Australian aesthetics theorist John Armstrong, in his “Note on Pronunciation”, an introduction page to his Love, Life, Goethe, gave the following well-rounded look at some of the issues, for non-German speakers, in attempts to render the name of Goethe: 
“Goethe’s name is something of a problem. At least for English speakers it raises a tricky issue of pronunciation.
Growing up in Glasgow I occasionally heard of a German writer called ‘Go-th’. It’s not a bad attempt and carries interesting resonances: Gothic architecture and the Goths—always linked with Vandals and Huns. My mother sometimes spoke of someone called ‘Gerter’ or ‘Gerder’—tentatively acknowledging the ‘otherness’ of the name. It has no ‘r’s’ in it so perhaps, by the perverse logic of a foreign tongue, you stress them precisely because they’re not there. It does sound quite Germanic. My mother is the perfect audience for the Dublin building-site joke:
A new construction worker is a bit confused and asks: ‘Joist? Girder?’ What’s the difference?’ The foreman patiently explains: ‘Joist wrote Uylsses, Girder wrote Faust.’
I was still following the construction-site usage when I happened to mention Goethe while teaching English as a foreign language. A German student assumed I was talking about someone he hadn’t heard of until I commented that, of course, the writer’s best-known work was Faust—at which point he said: ‘Oh, you mean “Goo-t’i”.’ (Say ‘oo’ while thinking intently about the letter ‘r’; the ‘I’ is weak, as in ‘it’).
‘Goo-t’i’ it is: in my opinion less pleasant than some of the mispronunciations. It’s an additional, if perhaps minor, obstacle to the reception of the man and his work. I still feel anxious when I mention his name to people who have heard of Faust but are entirely unsure who this ‘Goo’ person might be.
Although this is the settled modern pronunciation, it was not such a clear-cut matter in Goethe’s own day. There is a letter by the lovely Caroline Flachsland, who later knew Goethe very well, referring to ‘Gede’—a young poet (Goethe) all her friends were talking about. So they must have been pronouncing the great man’s name in a way that would upset today’s sophisticates.”
“I remember the first time I heard Goethe's name spoken aloud. I had a professor in college who taught modern American drama. The class was comprised of two elements: long, tedious theater anecdotes and unexplained references to figures, works, and ideas that no one in the class knew a thing about. I was that kind of generous spirit that assumed the teacher was not just an ostentatious prick but rather a caring soul who wanted us to learn something on our own, so when I heard a reference I did not understand, I would go look it up, thinking it might give me a better understanding of what we were talking about in class. It rarely did, though it at least let me know when he was just showing off, which was often.In 2010, American electrochemical engineer Libb Thims, published his million+ viewed “Smartest Person Ever”, a four-part video series on YouTube, Goethe ranking at #1, in the original making of which, Thims pronounced his name as “Goat-ah”, which quickly came under correction-attack in the threads; some examples of which are as follows: 
Anyhow, I can remember a whole class, I think it was on Eugene O'Neill, in which he dropped the name Faust into the discussion several times without ever telling us who this Faust was or what he had to do with Eugene O'Neill. And then he kept saying the name of the author, "Gurta". Gurta said this. Gurta said that, and so on. Well, I kept going to the library to look up Gurta, but there was no such author that I could discover. When I looked up Faust, I saw the name "Goethe," but the lack of an 'r' in the name threw me off the scent. I have a memory of it taking me a long time to put two and two together, but it might not have been that long. Memory is funny that way.”
“In addition to these mispronunciations: "Goat-a?" Who the hell is "Goat-a"? I think it's pronounced closer to something like "Gerta".”— turtlecasserole (2011), in reply to fecesfeces9, May
“No doubt Goethe can be called the smartest human ever. He is also one of the most influential. However, why can't the producer of this video learn how to pronounce this great genius' name. Please check out what the two little dots above the "o" do to the sound of that vowel.”— 1Victorinus (2011), Aug
|A 2013 Goethe pronunciation argument (Ѻ), in part three of the smartest person ever video.|
“Do you realize that you are pronouncing Goethe incorrectly?”
(add discussion)— 1anero (2012), May 2
“He's doing pretty well of course he can't pronounce "ö" properly at least he's trying.”— probaner92 (2012), in reply to 1anero, May 4
“GOAT-uh? What in the hell? Someone who claims to be in the position to discuss intelligence has no better sense than to pronounce Goethe as GOAT-uh?”— Chase Kimball (2012) (Ѻ)“It's not GU(R)-tuh, but Götuh: ö like u in "further.”— SellusionStar (2012) (Ѻ)“Goethe is pronounced something more like 'Ger-ta' than the pronunciation 'Go-the' used.”— aucourant (2016), comment (Ѻ) on “IQ 200+ | Smartest Person Ever (4 of 4)”, Jun 23