Even among scholars who don’t consider ‘Altaic languages’ to be a valid cladistic grouping it is common to think of these languages as at least typologically similar. Thus, it would be useful to sketch out what these common traits are, to what extent are they found in other (vaguely) neighbouring languages, and more importantly, what differences are found between them. A good starting point would be comparing the vowel systems.
The system usually considered most representative in naive approaches is that of Turkic, which contains in its most minimal form a perfect cube of 8 vowels distinguished by [+/-high][+/-front] and [+/-rounded]: /*a *ä *o *ö *u *ü *i *ï/.
However there are further complications, first of which being that there was also a closed *e with reflexes different from *ä in some languages, being written as i in Old Turkic and merging with i in Chuvash for example. Yet, some researchers have proposed that this phoneme is produced by an early soundlaw *ä: > *e:, however others claim that both pairs of *ä *ä: and *e *e: are distinguished.
Another proposed phoneme is *ë, a mid-back unrounded vowel, meant to explain the correspondence of Chuvash ï ~ Yakut ï ~ Other languages a, as well as cases of a appearing in some Old Turkic inscriptions, Chuvash, Yakut, Khalaj or Mongolian where other languages have ï. Nonetheless, these correspondences are imperfect and this phoneme is not accepted by all (or even most?) Turkologists.
And the third proposed addition is *ia which is introduced to explain cases of Chuvash Ću ~ Common Turkic *Ca ~ Mongolic *Cï where Ć symbolizes palatalization of the preceding consonant and u being the normal reflex of Turkic *a in Chuvash.  There is a theory that seeks to explain this phenomenon through regular breaking of *a: > *ia in Oghur, however sifting through Turkic comparative materials it is easy to find many exceptions to this rule.
Similar to Turkic, Proto-Mongolic vowel system is commonly denoted as /*a *e *o *ö *u *ü *i *ï/, however in most Mongolic languages ‘front’ and ‘back’ vowels are in fact distinguished by vowel height/pharyngealization/tongue root position. For example, the vowel system of Chakhar Mongolian is /ɑ ə ɔ o ʊ u i ɪ/, this is traditionally called ‘rotation’ with the idea that previously front vowels backed to their current position.
However as this backing is less phonetically plausible than the reverse direction, it is increasingly popular to posit that Proto-Mongolic was based on tongue root harmony. Yet this is difficult to reconcile with Middle Mongolian, where the spelling of *ü implies palatality in Uighur script (ui), ‘Phags-pa (eeu) and Sino-Mongolian (using characters with /y/ and /ju/ medials although plain /u/ is also attested). Another important bit of data here is Khitan, which predates Proto-Mongolic as commonly reconstructed, that has both tongue root and (secondary) palatality contrasts according to Ōtake Masami.
And another smaller question is whether *ï is a phoneme or merely an allophone of *i occuring in ‘back’ vowel words. Even though no language preserves the distinction as phonemic, it seems more convenient to reconstruct due to cases like *čïkïn ‘ear’ and *čïkï- ‘to squeeze’ appearing with uvulars in Middle Mongol and *čïsun ‘blood’ and *(s)isün ‘soot’ having the same suffix but belonging to different classes.
Diphthongoids, including double vowels, are in Mongolic in some cases contractions of earlier *-VCV- sequences as I’ve explained earlier, currently I know of no reason to presume there were any such sequences of different origin. (I may cover supersegmentals in another post)
In a sense, Mongolic vocalism stands between Turkic and Tungusic, which should not be surprising.
Unlike Mongolic, I have no doubt at all that Tungusic vocalism is based on tongue root position rather than palatality. As a maximal reconstruction I would propose:/*a *ə *o *ȯ *ụ *u *ü *ị *i/, where dots below and above represent retracted and advanced tongue root respectively (caveat: some sources use the dots with the exact opposite meaning) and *ü is an actual [y] or at least [ʉ].
Here we have a similar situation as with Mongolic *ï, retracted high vowels *ụ and *ị are likely merely allophones of *u and *i, recessively taking on retraction of other vowels in the word. In cases where there are only high vowels, they are advanced in monosyllabics and retracted otherwise, at least this is how the theory goes.
*ȯ is also not accepted by all sources, but there’s a number of fairly solid correspondence sets Evenki u ~ Even ө ~ Orok o ~ Nanai u that seem most elegantly explained through such a reconstruction.
The only diphthong accepted by all sources is *ia which has a unique reflex in all branches. Some reconstructions also have *iə, *ua, *uə, *ui and perhaps *ai. These are usually best preserved in Nanaic.
Late Middle Korean vowel system is often transcribed as /a e o wo u wu i/ supposedly representing [a ə ʌ o ɨ u i], with vowel harmony grouping low(er) a o and wo against higher e u wu and i[a ə ʌ o ɨ u i being neutral as also suggested by some analysis of Mongolic and Tungusic. However some scholars have suggested that Korean vowel harmony is a Middle Korean innovation.
Korean also features many diphthongs with on- and offglides y and w. Alexander Francis Ratte in his dissertation on Proto-Korean-Japanese seems to consider all of these secondary (deriving from consonants and contractions of multiple syllables) except for ye which he derives from Proto-Korean *e. 
It seems well accepted that Proto-Japanese vowels are /*a *e *ə *o *u *i/ , and that the more complex system of Old Japanese vowels arose through contraction of earlier diphthongs (see for example Proto-Japonic *e and *o in Eastern Old Japanese).
Old Japanese also features a limitation on vowels co-occuring in roots that is sometimes compared with vowel harmony. According to Ratte, it amounts to *ə never co-occuring with *a, *o and *u, indeed this doesn’t seem to have much in common with other cases of Eurasic vowel harmony and he explains it through an earlier assimilation of *ə around these vowels, which does appear to be a priori likelier.
The Uralic system consists of /*a *ä *e *ë *o *u *ü *i/, frontness being the distinguishing feature between umlauted pairs and with *ï sometimes added or *ë sometimes denied.
Despite incompleteness, this seems most similar to Turkic vocalism among neighboring languages (excluding traditional reconstructions of Mongolic). Perhaps it could be speculated that Turkic vocalism is itself a product of Uralic (or specifically Samoyedic) influence.
Proto-Indo-European possesses a minimalist inventory of /e *o *ē *ō/, marginally and controversially *a ā, and *i *u as syllabic glides. Considering ablaut alternations, it is probable that it previously had even fewer vowels (and if glides don’t derive from breaking of some sort).
If Indo-European is related to some of the Eastern languages (Indo-Uralic being a popular hypothesis), clearly it would have to had gone through a complete collapse of its vowel system, perhaps under the influence of Caucasian languages (although Chirikba reconstructs pre-West Caucasian with *ö and *ü).
Even though early Turks partly derive genetically from Andronovo culture populations, who presumably spoke Indo-Iranian, their reduced vowel system hasn’t left a recognizable trace.
I haven’t read the literature on comparative Yeniseian phonology, but from the bits and pieces I’ve seen, it seem that Proto-Yeniseian is taken to have the same inventory as Ket: /a *e *ə *o *ɨ *u *i/ with /ə ɨ/ possibly being /ɤ *ɯ/. 
I am hoping that investigation of the Dene-Yeniseian connection will offer further glimpses into the past of Yeniseian itself, and hopefully into the deep prehistory of Siberia.
For Proto-Yukaghir Nikolaeva reconstructs /*a *e *o *ö *u *ü *i *ï (her *y)/, a full cubic system based on frontness. However this doesn’t describe the situation as found in neither Tundra nor Kolyma Yukaghir.
This back vocalic *ï surfaces as normal i, broken ya or a as well as lowered e, which is to say, in every way except as back vocalic. Nikolaeva also mentions that “In the modern languages u is harmonically back in only a few words, mostly from Tundra Yukaghir. Such words are normally fairly recent borrowings from Tungus”, which necessitates reconstructing *ü, which reflected as a back vowel in both languages but belongs to the harmonically “front” class.
Unlike *ü, *e and *ö are found as actual front vowels in both Tundra and Kolyma, yet Nikolaeva mentions that *e was written with a or o in the earliest attestation, and while she interprets this as indicating a lower /æ/, it suggests a central /ə/ to me.
Fortescue gives /*a *æ *e *ə *o *u *i/ for Proto-Chukotko-Kamchatkan with recessive vowel harmony (umlaut?) based on ‘dominant’ *a *e *o against ‘recessive’ *æ *i *u and *ə being a reduced vowel acting different from full vowels.
Proto-Eskimo-Aleut had /*a *ə *u *i/, the simplest (outside of Proto-Indo-European) of the vowel systems discussed here. Perhaps this is a result of a collapse of a Chukotkan-like system as happened in Kerek and Alutor, but this is baseless speculation.
Nivkh possesses an ideal height harmony system: /*a *ə *o *u *e *i/ which is reflected basically as such in all dialects, although actual harmonic alternations themselves have been lost in most cases.
Most Ainu dialects apparently have simple vocalism, /*a *e *o *u *i/, but in a 1992 paper Vovin reconstructs a more complicated system with four height levels, front rounded vowels, a back unrounded *ï and two low vowels. I do not know how accepted these ideas are among other scholars.
Old Chinese is reconstructed with /a ə o u e i/. Marc Miyake presents an interesting idea that Old Chinese had an opposition between high and low vowels and that pharyngealization in late Old Chinese was caused by low vowels in presyllables which were lost.
Tangut also had a tense/lax distinction which might also be related to the Chinese situation, but Tangut materials are too opaque for me to grok.
By now, everyone interested in this type of stuff heard of the idea of advanced tongue root harmony for Proto-Mongolic . Yet, it’s acceptance hasn’t been universal. In a relatively recent paper Janhunen defends original palatal harmony not only in Mongolic but Tungusic, Korean and Nivkh by noting
There are, indeed, good arguments, both internal and external, for assuming that vowel rotation was a process that was active in Northeast Asia, and that started relatively late, reaching different languages and language families in different times..
To me, it seems more inherently more plausible that a rotation happened in reverse direction, and quite unacceptable to overrule the testimony of attested language by what I suspect to be a projection archetypal Ural-Altaic languages based on familiarity. There must be something beyond that, but I am yet to hear these arguments.
 Some sources would reconstruct other *iV diphthongs such as *io or *iu, but if diagnostic palatalization in Chuvash appears in these positions as well, perhaps we would be better off reconstructing palatalized consonants.
 Even if Japanese and Korean may not form a valid genetic grouping, Ratte’s work seems to grasp at something real, presumably due to shared history between Korean and Japanese, just as comparisons of Turkic, Mongolic and Tungusic offer insights into their prehistory that would be impossible otherwise.
 This is Vajda’s analysis. There are alternative takes, covered by Vajda here, but his appears most convincing.
 One of the earliest appearances of this theory is this paper, written by an author with a primary focus on phonology.