Michael Everson correctly identifies a number of reasons to advocate the disunification of the Latin letters beta, theta, and chi from their Greek versions. If this happened, as IPA symbols we would use the Latin versions rather than the Greek ones.
He quotes briefly, without identifying the source, from the IPA 1949 Principles booklet. Here, more fully, is what is says there (The Principles of the International Phonetic Association, pages 1-2). Although unattributed, these are clearly Daniel Jones’s words.
Note the very clear intention to treat IPA θ (vertical) as distinct from Greek theta (typically oblique). Greek letters are to be incorporated into the IPA only as roman [sic] adaptations.
As Jones says, Greek theta has an alternative form, ϑ. This is encoded at U+03D1, whereas ordinary θ is at U+03B8.
In English printed texts that mix the Latin and Greek scripts, the Greek letters are typically oblique, the Latin ones upright. The purpose is to distinguish clearly between the two scripts (whereas the IPA wants everything in the same script). Here is an example, from Abbott and Mansfield’s Primer of Greek Grammar (my copy printed in 1949).
I think disunification of Latin and Greek beta, theta, chi would be a good thing.
An existing disunification that might be thought surprising is that of the IPA symbol for a voiced velar plosive, ɡ, U+0261, from ordinary lower-case g, U+0067. In many fonts there is no difference in the appearance of these two; in other fonts there is, e.g. in Times New Roman ɡ g (which I hope shows up properly in your browser). The IPA is on record as declaring that the two symbol shapes are equivalent and interchangeable. Nevertheless many phoneticians persist in treating them as distinct, which justifies Unicode’s disunification.
It is worth noting that a number of obsolete, derecognized former IPA symbols are located in the Unicode block Latin Extended-B. They include ƍ ƞ ƪ ƫ ƺ ƾ ƻ. This is also where we find upper-case versions of certain IPA symbols. These might be used in orthographies, though not in phonetic texts as such: Ɔ Ə Ɛ Ɣ Ɯ Ɵ Ʊ Ʌ. I have sometimes had to correct careless authors who used them in place of the lower-case phonetic symbols.
Another difficult area is that of letters with diacritics. It is possible to encode any such letter by using the base form plus one (or more) of the Combining Diacritical Marks provided in Unicode 0300–036F. However, doing so puts you at the mercy of the designers of fonts, browsers and word processing software, who may or may not have done the necessary work to make diacritics line up correctly above, below, or through the base letter. For “accented” letters used in orthographies Unicode provides separate encoding, as for example in the case of precomposed á ê ï õ ù ă ē į ő ů ç đ ġ ķ ň. However no precomposed combinations are provided for explicitly phonetic use. Obviously, since the range of possible combinations is potentially enormous we cannot expect to have many of these; but it would certainly be convenient to have precomposed versions of the symbols for the French nasalized vowels (blog, 15 July), ɑ̃ ɛ̃ ɔ̃ œ̃, which are abundantly attested in printed texts.