If you want to use UTF-8 characters for the phonetic symbols, you have to teach them to LaTeX, because the tipa package doesn't provide support for this. Alternatively, you can input them with the shorthands provided by the package.
\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{tipa}
\usepackage{newunicodechar}
\newunicodechar{ˈ}{"}
\newunicodechar{ə}{@}
\newunicodechar{ɛ}{E}
\newunicodechar{ɪ}{I}
\begin{document}
\textipa{/fəˈnɛtɪk/}
\textipa{/f@"nEtIk/}
\end{document}

How to extend the set of supported characters? Let's see, for instance,
ˌjuːnɪˈvɜːsɪtɪ
I added \textipa{/ˌjuːnɪˈvɜːsɪtɪ/} to the example above and, upon running pdflatex, I saw the errors
Unicode char ˌ (U+2CC) not set up for use with LaTeX
Unicode char ː (U+2D0) not set up for use with LaTeX
Unicode char ɜ (U+25C) not set up for use with LaTeX
Then I went to the manual of tipa, Appendix A Annotated List of TIPA Symbols and looked for the characters, just like I did for the previous ones. The last one appears like in the image below

so I know what shorthand to put in the second argument to \newunicodechar.
Here is the full example:
\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{tipa}
\usepackage{newunicodechar}
\newunicodechar{ˈ}{"}
\newunicodechar{ə}{@}
\newunicodechar{ɛ}{E}
\newunicodechar{ɪ}{I}
\newunicodechar{ˌ}{""}
\newunicodechar{ː}{:}
\newunicodechar{ɜ}{3}
\begin{document}
\textipa{/ˌjuːnɪˈvɜːsɪtɪ/}
\textipa{/fəˈnɛtɪk/}
\textipa{/f@"nEtIk/}
\end{document}

Instead of \newunicodechar you could use
\DeclareUnicodeCharacter{025C}{3}
instead of \newunicodechar{ɜ}{3}, but I find it more convenient to have the “real” character instead of its code point.
tipapackage? – egreg Mar 20 '16 at 17:28