The problem you are having is due to the fact that tipa redefines basic letters within its IPA environment, which is triggered by \textipa{...}, {\tipaencoding...}, or \begin{IPA}...\end{IPA} (see section 3.2.1 of the tipa manual). As you can see on p. 44 of the manual, a capital V within the IPA environment is typeset as ʋ:

This capital letter problem is Q5 in the FAQ section of the manual (pp. 63-64). Here are three ways around it.
1. Use a Unicode IPA font and XeLaTeX/LuaLaTeX instead of tipa
My top recommendation is to avoid using tipa for IPA transcription for the reasons outlined at Typesetting phonetic symbols: Unicode or tipa?.
Here is a MWE that can be compiled using XeLaTeX or LuaLaTeX (and ShareLaTeX does allow you to pick which compiler to use). I typed all the diacritics directly into my TeX using an IPA keyboard, but you could use an online IPA picker or copy and paste from some other source.
\documentclass{article}
\usepackage[charter]{mathdesign} % loads a math font to match Charis SIL
\usepackage[no-math]{fontspec} % package responsible for font selection in XeLaTeX/LuaLaTeX
\setmainfont{Charis SIL} % sets the document font to Charis SIL, which you would need to have on your system or loaded into ShareLaTeX
\begin{document}
The equivalence of the medial V̄C and V̆CC sequences in the classical languages has long been recognized. It may in fact be rather more broadly stated than we have done so far. V̄C may be restated as V̄C$_0$, where C$_0$ = any number of consonants, including zero (e.g.\ Latin \emph{diḗi, relā́tus, redā́ctus, redḗmptus}); and V̆CC as V̆C$_2$, where C$_2$ = at least 2 consonants.
\end{document}

2. Use \={} and \u{} outside of the tipa IPA environment
Regardless of whether you decide to use a Unicode font with XeLaTeX/LuaLaTeX or to continue using the palatino and tipa packages with pdfLaTeX, you can use the standard LaTeX macros \={} and \u{} for the macron and breve, as long as you do it outside of the tipa IPA environment. This is what you showed in your MWE.
3. Use the \* to access the default C and V within the tipa IPA environment
As discussed in section 3.2.4 of the tipa manual, \* may be used before some letters (including all capital letters) within the IPA environment to escape the environment's redefinitions and print the normal letter.
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{tipa}
\begin{document}
The equivalence of the medial \textipa{={*V}*C} and \textipa{\u{*V}*C*C} sequences in the classical languages has long been recognized. It may in fact be rather more broadly stated than we have done so far. \textipa{={*V}*C} may be restated as \textipa{={*V}*C}$_0$, where \textipa{*C}$_0$ = any number of consonants, including zero (e.g.\ Latin \textsl{\textipa{di'{={e}}i, rel'{={a}}tus, red'{={a}}ctus, red'{={a}}mptus}}); and \textipa{\u{*V}*C*C} as \textipa{\u{*V}*C}$_2$, where \textipa{*C}$_2$ = at least 2 consonants.
\end{document}

Note that this works with the default Computer Modern font, but it is not possible to use tipa and get IPA output that matches the Palatino font you had in your MWE (below, the Latin words appear in tipa Computer Modern lookalike font, and they cannot be slanted because Palatino doesn't have a slant style).
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{mathpazo} % for Palatino-looking math
\usepackage{palatino}
\usepackage{tipa}
\begin{document}
The equivalence of the medial \textipa{={*V}*C} and \textipa{\u{*V}*C*C} sequences in the classical languages has long been recognized. It may in fact be rather more broadly stated than we have done so far. \textipa{={*V}*C} may be restated as \textipa{={*V}*C}$_0$, where \textipa{*C}$_0$ = any number of consonants, including zero (e.g.\ Latin \textsl{\textipa{di'{={e}}i, rel'{={a}}tus, red'{={a}}ctus, red'{={a}}mptus}}); and \textipa{\u{*V}*C*C} as \textipa{\u{*V}*C}$_2$, where \textipa{*C}$_2$ = at least 2 consonants.
\end{document}

\={V}and\u{V}should work (not usingtipacommands) – egreg May 16 '16 at 16:01tipatry\*, i.e.\={\*V}– Au101 May 16 '16 at 16:59\={V}CCas argument to\textipa? They're not IPA symbols. – egreg May 16 '16 at 17:40\u{\={\*V}}and even\'{\u{\={\*V}}}both will be produced perfectly bytipabut will not work outside it with Computer Modern. However, although they are not IPA symbols, they are phonology symbols and I can certainly imagine cases where you might want to do something like\u{\*V}X $\to$ \={\*V}(a mostly fictional, highly simplistic, compensatory lengthening process based on English 'night' – Au101 May 16 '16 at 18:24