Drawing arrow heads without the tails in TikZ?

You can use path decorations:

\documentclass{article}
\usepackage{tikz}

\usetikzlibrary{decorations.markings}

\begin{document}

\begin{tikzpicture}[decoration={
  markings,
  mark=at position 0.2 with {\arrow{<}}}
]
  \draw[postaction={decorate}] (0,0) ellipse (1 and 2);
\end{tikzpicture}

\end{document}

Tested with TikZ 3.0:

• load the arrows.meta library instead of arrows
• add draw opacity=0 as option to \draw, but not to the arrow, such as >={LaTeX[]}

The tail will be completely opaque, i.e. will not be shown, only the head will be visible.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{arrows.meta}
\begin{document}
\begin{tikzpicture}
  \draw[->,>={LaTeX[]},draw opacity=0] (0,2) arc (90:45:1 and 2);
  %\draw (0,0) ellipse (1 and 2);% commented out to see only the tip
\end{tikzpicture}
\end{document}

(Tested with TikZ 3.0.1a)

You can use the tips key to configure drawing of arrow tips. The initial value of the tips key is on draw, so initially, arrow tips are only drawn when the path itself is drawn, too. But you can set the tips key to true to draw arrow tips even for undrawn paths.

\path[tips, ->] (0,2) arc (90:45:1 and 2);
\draw (0,0) ellipse (1 and 2);

See Chapter 16.2 (Where and When Arrow Tips Are Placed) of the TikZ and PGF Manual.