Is Missouri called /mi.'zuɹ.ə/ outside of Missouri?

Having grown up in a nearby metropolis (Tulsa, OK), and married to a person who grew up in one on the opposite side of that state (Cinncinnati, OH), I can tell you that outside of Missouri, almost everyone in neighboring states pronounces it with a long-E sound.

I had relatives in the extreme Southwestern county (Macdonald) of Missouri, and they pronounced it with the "uh" sound at the end instead. As a result, I learned to code-switch: Inside Missouri it is "Missouruh", and outside "Missouree". So I'm fairly certain that pronunciation is just something residents use locally to identify fellow residents (I've observed this phenomenon in more localities than I can mention. Most prominently New Orleans, which residents pronounce more like "Nawlins").

As far as dialect goes, Missouri is an interesting case. You could simplify and just say the state speaks American Midland, which is probably the closest live dialect to "Standard American English". However, the truth is not quite so boring. St. Louis has its own dialect, but otherwise the state exists on the border between the North Midland and South Midland dialect. My SW Missouri relatives definitely were in the South Midland camp.

University of Missouri English professor Donald Lance made an impressively complete study of the high vowel vs. schwa issue. It turns out that its use in the confines of Missouri is even more complex than outside (so thankfully you didn't ask about inside the state). But it does appear that the original introduction of that pronunciation is most likely a feature of the South Midland dialect area. As of the 1960's it was in fact still pronounced that way by some folk outside the state.

However, younger people (I'm guessing those growing up with access to national media via radio and TV), largely quit pronouncing it that way.

So if you find yourself a non-Missourian South Midland speaker born before 1930, there should be a good chance you will hear "Missourah".

From Wikipedia:

The state is named for the Missouri River, which was named after the indigenous Missouri Indians, a Siouan-language tribe. They were called the ouemessourita (wimihsoorita), meaning "those who have dugout canoes", by the Miami-Illinois language speakers. As the Illini were the first natives encountered by Europeans in the region, the latter adopted the Illini name for the Missouri people.

While many American states have names that its natives and non-natives pronounce dissimilarly, Missouri is the only one whose name is pronounced differently even just among its present-day natives — the two most common pronunciations being /məˈzɜri/ and /məˈzɜrə/. This situation of differing pronunciations has existed since the late 1600s. Further pronunciations also exist in Missouri or elsewhere in the United States, involving the realization of the first syllable as either /mə/ or /mɪ/; the stressed second syllable as either /ˈzɜr/ or /ˈzʊər/; the third syllable as /i/, /ə/, centralized /ɪ/ ([ɪ̈ ]), or even ∅ (in other words, a non-existent third syllable); and the phoneme /r/ as either of two allophones: [ɹ] or [ɻ]. Any combination of these phonetic realizations may be observed coming from speakers of American English.

Politicians often employ multiple pronunciations, even during a single speech, to appeal to a greater number of listeners. Often, "eye dialect" spellings of the state's name, such as "Missour-ee" or "Missour-uh," are used informally to phonetically distinguish pronunciations.

Personally, I use /məˈzɜri/ - though it's hard to say where I picked up the pronunciation from (North-Eastern US, Texas, or South Florida are most likely). The most common pronunciations I hear are /məˈzɜri/ and /məˈzʊəri/ (the later, for example is used by a friend of mine raised in Canada as a child and moved to Atlanta, Georgia in her teens)

Edit: First paragraph from the wikipedia article added to the quote and the link is now provided.

I'm from rural Southeast Missouri, Perry County to be exact. I have never heard anyone in either Southeast Missouri or in St. Louis ever use the pronunciation /mi.'zuɹ.ə/. I've always assumed this stereotype of Missourians using /ə/ at the end of the name was either a joke or myth held in popular belief by non-Missourians.

Having lived in New York and Southern California, I will vouch for at least SoCal (what does it mean to be a New Yorker, anyway?) that we pronounce it /miz.'uɹ.i/ , and I have not encountered a person from outside the general vicinity of Missouri who pronounces it as those do who live within the said general vicinity.
Here's a site with a bunch of sources which argues for regional pronunciation patterns which would support OP's idea. As an additional lead for your research, note the recent BusinessInsider article about regional pronunciations of individual words, which you have perhaps already seen.
I can't speak to the point on the South. Am quite curious as well.