How to put an accent mark on a barred i (U+0268) without the tittle/dot

This seems quite enough for the standard sizes.

\documentclass{article}
\usepackage{tipa}
\usepackage{pict2e,picture}
\usepackage{color}

\makeatletter
\newcommand{\textdotlessbari}[1]{%
  \begingroup
  \settowidth\dimen@{\textbari}%
  \makebox[\dimen@]{%
    \sbox\z@{\i}%
    \begin{picture}(0,\ht\z@)
    \roundcap
    \linethickness{0.055ex}
    \put(0.05\wd\z@,0.55\ht\z@){\line(1,0){0.9\wd\z@}}
    \end{picture}%
    #1\i
  }%
  \endgroup
}
\DeclareRobustCommand{\textbariacute}{\textdotlessbari{\'}}
\DeclareRobustCommand{\textbarigrave}{\textdotlessbari{\`}}
\makeatother

\begin{document}

--\textbari--

--\textbariacute--

\leavevmode
\rlap{\textcolor{red}{\textbari}}\textbariacute
\rlap{\textcolor{red}{\textbariacute}}\textbari

\textbariacute\textbarigrave\textdotlessbari{\~}

\Huge
--\textbari--

--\textbariacute--

\leavevmode
\rlap{\textcolor{red}{\textbari}}\textbariacute
\rlap{\textcolor{red}{\textbariacute}}\textbari

\textbariacute\textbarigrave\textdotlessbari{\~}

\end{document}

The third line shows the characters superimposed to one another in different color and the differences are very small to be really noticeable.

Use \rlap{-}\'\i or \leavevmode\rlap{-}\'\i if that symbol start a paragraph. Example:

\documentclass[12pt]{article}
\begin{document}
\leavevmode\rlap{-}\'\i p  ip  \rlap{-}\'\i p ip
\end{document}