Update
Now there is a MWE in the question and I understand that you need the change (only?) for your custom sectionning command \Article. You can simple adapt the solution from the original answer below:
\documentclass{scrartcl}
\linespread{1.1}
\setlength{\parindent}{0.5cm}
\usepackage{lipsum}% only for dummy text
\DeclareNewSectionCommand[
style=section,
level=\subsubsectionnumdepth,
beforeskip=3.25ex plus 1ex minus .2ex,
afterskip= 1.5ex plus .2ex,
indent=0pt,
font=\mdseries\slshape,
%tocstyle=gobble,
tocstyle=default,
toclevel=\subsubsectiontocdepth,
tocindent=4em,
tocnumwidth=0pt% <- changed to 0pt, tocdynnumwidth and tocentrynumberformat are removed
]{Article}
\renewcommand\Articleformat{Article~\theArticle\autodot\enskip}
\renewcommand*\addArticletocentry[2]{%
\addtocentrydefault{Article}{}
{\IfArgIsEmpty{#1}{}{Article~\makebox[1.5em][l]{#1\protect\autodot}}#2}%
}% maybe you have to adjust the 1.5em -> run twice to see the change in TOC
\renewcommand\sectionlinesformat[4]{%
\ifstr{#1}{Article}
{\raggedright\textbf{\upshape#3}#4.}
{#3#4}%
}
\begin{document}
\tableofcontents
\lipsum[1-2]
\Article{Short title}
\lipsum[3]
\Article{Very long article title that (on the Table of Contents) should start on the beginning of the line instead of at the beginning of the word Article}
\lipsum[4]
\end{document}
Result:

But maybe you want:

\DeclareNewSectionCommand[
style=section,
level=\subsubsectionnumdepth,
beforeskip=3.25ex plus 1ex minus .2ex,
afterskip= 1.5ex plus .2ex,
indent=0pt,
font=\mdseries\slshape,
%tocstyle=gobble,
tocstyle=default,
toclevel=\subsubsectiontocdepth,
tocindent=0pt,% <- changed
tocnumwidth=0pt% <- changed to 0pt, tocdynnumwidth and tocentrynumberformat are removed
]{Article}
\renewcommand\Articleformat{Article~\theArticle\autodot\enskip}
\renewcommand*\addArticletocentry[2]{% <- definition changed
\addtocentrydefault{Article}{}
{\IfArgIsEmpty{#1}{}{\hspace{4em}Article~\makebox[1.5em][l]{#1\protect\autodot}}#2}%
}% maybe you have to adjust the 1.5em -> run twice to see the change in TOC
Or:

\DeclareNewSectionCommand[
style=section,
level=\subsubsectionnumdepth,
beforeskip=3.25ex plus 1ex minus .2ex,
afterskip= 1.5ex plus .2ex,
indent=0pt,
font=\mdseries\slshape,
%tocstyle=gobble,
tocstyle=default,
toclevel=\subsubsectiontocdepth,
tocindent=0pt,
tocnumwidth=0pt
]{Article}
\renewcommand\Articleformat{Article~\theArticle\autodot\enskip}
\renewcommand*\addArticletocentry[2]{% <-definition changed
\addtocentrydefault{Article}{}
{\IfArgIsEmpty{#1}{}{Article~\makebox[1.5em][l]{#1\protect\autodot}}#2}%
}% maybe you have to adjust the 1.5em -> run twice to see the change in TOC
Original answer
Unfortunaly there is no MWE in the question, so its only a guess:
\documentclass{scrartcl}
\RedeclareSectionCommands[%
tocnumwidth=0pt%
]{section,subsection,subsubsection,paragraph,subparagraph}
\renewcommand*\addsectiontocentry[2]{%
\addtocentrydefault{section}{}
{\IfArgIsEmpty{#1}{}{\makebox[1.5em][l]{#1\protect\autodot}}#2}%
}
\renewcommand*\addsubsectiontocentry[2]{%
\addtocentrydefault{subsection}{}
{\IfArgIsEmpty{#1}{}{\makebox[2.3em][l]{#1\protect\autodot}}#2}%
}
\renewcommand*\addsubsubsectiontocentry[2]{%
\addtocentrydefault{subsubsection}{}
{\IfArgIsEmpty{#1}{}{\makebox[3.8em][l]{#1\protect\autodot}}#2}%
}
\renewcommand*\addparagraphtocentry[2]{%
\addtocentrydefault{paragraph}{}
{\IfArgIsEmpty{#1}{}{\makebox[4.1em][l]{#1\protect\autodot}}#2}%
}
\renewcommand*\addsubparagraphtocentry[2]{%
\addtocentrydefault{subparagraph}{}
{\IfArgIsEmpty{#1}{}{\makebox[5em][l]{#1\protect\autodot}}#2}%
}
\usepackage{lipsum}% only for dummy text
\begin{document}
\tableofcontents
\clearpage
\addsec{Lorem ipsum dolor sit amet, consectetuer adipiscing elit.
Ut purus elit, vestibulum ut, placerat ac, adipiscing vitae, felis.}
\lipsum[1-5]
\section{Lorem ipsum dolor sit amet, consectetuer adipiscing elit.
Ut purus elit, vestibulum ut, placerat ac, adipiscing vitae, felis.}
\lipsum[6]
\subsection{Lorem ipsum dolor sit amet, consectetuer adipiscing elit.
Ut purus elit, vestibulum ut, placerat ac, adipiscing vitae, felis.}
\lipsum[7-10]
\subsection{Lorem ipsum dolor sit amet, consectetuer adipiscing elit.}
\lipsum[11-15]
\end{document}
