Newlines after section headings in amsart

You can redefine \subsubsection appropriately; here's the original definition in amsart.cls

\def\subsubsection{\@startsection{subsubsection}{3}%
  \z@{.5\linespacing\@plus.7\linespacing}{-.5em}%
  {\normalfont\itshape}}

and you can redefine it using something like this:

\documentclass[9pt]{amsart}
\usepackage[parfill]{parskip}    

\makeatletter
\def\subsubsection{\@startsection{subsubsection}{3}%
  \z@{.5\linespacing\@plus.7\linespacing}{.1\linespacing}%
  {\normalfont\itshape}}
\makeatother

\begin{document}
\subsubsection*{Animals}
Rabbit horse monkey cow
\end{document}

Another option, not requiring knowing the inner definition of \subsubsection, is to use the titlesec package:

\documentclass[9pt]{amsart}
\usepackage[parfill]{parskip}
\usepackage{titlesec}

\titleformat{\subsubsection}
  {\normalfont\normalsize\itshape}{\thesubsubsection}{1em}{}
\titlespacing*{\subsubsection}{0pt}{3.25ex plus 1ex minus .2ex}{0ex plus .2ex}

\begin{document}
\subsubsection*{Animals}
Rabbit horse monkey cow
\end{document}

However, if you are writing a paper for a submission to an AMS journal, pay attention to the following warning from barbara beeton regarding the use of titlesec with the AMS document classes:

i can't recommend titlesec because it's an option we don't permit when a paper is submitted to an ams journal, so we've never tested it. (for the same reason, i can't "unrecommend" it either. for whatever it's worth, it has never been identified as causing failures with amsart; just "not tested here".)

A simple workaround not requiring any modification.

If you directly put \\ then it tells you there is no line here to end, but if you put any character before \\ it will allow you to create a new line.

So my workaround is put a ~ right after the \subsubsection{tittle}, and \\ after ~, the ~ create an invisible space after your title, and which enables you to create a new line for your visible paragraphs.

The simplest way would be to give subsubsection some invisible text before the newline:

subsubsection*{Animals}~\\Rabbit horse monkey cow

You may choose \mbox{} instead of ~.