With the current TU default encoding, \textasteriskcentered is mapped to U+2217 ASTERISK OPERATOR which Times New Roman happens to miss.
The old EU1/EU2 encodings are based on xunicode that mapped \textasteriskcentered to U+002A ASTERISK, which was the wrong thing to do, since the output when used as a footnote marker is definitely bad:

Compare with the output we get with TeX Gyre Termes

If I add
\UndeclareTextCommand{\textasteriskcentered}{TU}
\DeclareRobustCommand{\textasteriskcentered}{%
\iffontchar\font"2217 \char"2217 \else\loweredasterisk\fi
}
\newcommand\loweredasterisk{\raisebox{-.5ex}{*}}
to the preamble, I get, with Times New Roman,

which is still not very good, but definitely better than the previous output.
An altogether different solution: use TeX Gyre Termes for the symbol. You could also add code for coping with different font families.
\documentclass{book}
\usepackage{fontspec}
\usepackage{etoolbox}
\usepackage[
main=english,
]{babel}
\setmainfont{Times New Roman}
\newfontfamily{\termes}{TeX Gyre Termes}% for the asterisk
\makeatletter
\patchcmd{\@fnsymbol}
{\textasteriskcentered}
{{\termes\textasteriskcentered}}
{}{\ddt}
\makeatother
\begin{document}
\renewcommand*{\thefootnote}{\fnsymbol{footnote}}
\chapter{Symbolic footnotes and Times New Roman or Arial}
First footnote\footnote{Hello World!}\\
Second footnote\footnote{Hello Again!}\\
Third footnote\footnote{Heeeeello Agaiiiiiinn!}
\bigskip
Question(s):
When I use TNR or Arial font for example, the first footnotemark symbol (asterisk) is missing. Why?
(Note: Using Latin Modern, the output is fine.)
\end{document}

\DeclareTextSymbol{\textasteriskcentered}\UnicodeEncodingName{"002A}-- This should be fixed by the LaTeX-Team