What LaTeX macros cannot accept a group as an argument?

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All TeX macros take brace groups around their arguments, which may be omitted if the argument is a single token, although in the case of delimited arguments that group may not be where you mean.

\def\foo#1#2{...}

then \foo12, \foo 1 2, \foo{1}{2}, \foo{1} {2} are all equivalent.

\def\foo(#1,#2){...}

then \foo(1,2), \foo({1},{2}), \foo(1,{2}), are all equivalent.

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TeX primitives do not have optional braces in this way.

\hbox{a} the braces are mandatory \hbox a is a syntax error, however the brace can be implicit \hbox\bgroup a} is OK.

\hskip 5pt takes no braces, \hskip{5pt} is a syntax error

\def\foo{zzz} must have no brace around the token being defined,

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Some commands documented as taking arguments do not, so for example \section is a macro which takes no arguments it just looks ahead for * or [ calling appropriate macros in each case, so

\section*[short]{l} could be written as \section*[{short}] l but not \section{*}[short]{l}

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Some packages redefine ^ and _ to be macros and take arguments with optional brace groups as above, but the primitive behaviour is "interesting" as tex expands looking for possibly implicit braces so for example x^\mathrm x is x^{\mathrm{x}} and x^\frac12 is x^{\frac{1}{2}} as the first level expansions of \mathrm and \frac provide the {} group needed for the superscript.

This is not a complete answer but anything that uses \@ifnextchar can cause problems. I borrowed \CheckMinus from this answer.

\documentclass{article}
\makeatletter
\def\CheckMinus{\@ifnextchar-{got a minus}{no minus}}
\makeatother
\begin{document}
$\left(1\right)$ % works
%$\left{(}1\right{)}$ % doesn't work

\CheckMinus-

\CheckMinus{-}

\end{document}

Quite a few TeX "primitives" take arguments that must not be delimited in a group.

• Text-mode and math-mode kerning commands: \kern and \mkern, which take 1 argument each.

  a\kern1em a
  $a\mkern18mu a$
  

  are both ok, but a\kern{1em} a and $a\mkern{18mu} a$ are not.

• The commands \raise and \lower, which take 2 arguments each, the first of which must be a length variable and the second of which must be a (TeX) box.

  a\raise1ex\hbox{c}
  

  is ok, but a\raise{1ex}\hbox{c}, a\raise1ex{\hbox{c}}, and a\raise{1ex}{\hbox{c}} are not.

  The LaTeX counterpart to a\raise1ex\hbox{c} is \raisebox{1ex}{c}, which also takes two arguments. Clearly, with \raisebox is allowed -- and, depending on the arguments, may in fact be required.