Recommended solution
There's no really simple way to do this elegantly using natbib, since the .bst files that format the bibliography entries either produce the fields or not. There's no way to selectively turn them on or off. For this reason, I would recommend that instead of using natbib you use biblatex to manage your bibliography, since this will allow you to have direct control over whether to produce them or not.
See
and then the solution in the question that Joseph linked to:
will work.
One minor problem with this solution however, is that there are no biblatex styles that mimic plainnat exactly, mainly because most Author-Year systems put the year right after the author, and not right at the end like plainnat does. If you have any flexibility in choosing a style, you can choose one of the standard biblatex authoryear styles, the apa style, or the biblatex-chicago package with the authordate style. Here's a minimal example using the APA style:
% !BIB TS-program = biber
\RequirePackage{filecontents}
\begin{filecontents*}{\jobname.bib}
@ARTICLE{AD.Smith2001,
author = {Arthur D. Smith},
title = {A simple model of LaTeX References.},
journal = {Journal of LaTeX},
year = {2001},
volume = {100},
pages = {1--10},
number = {3},
keywords = {LaTeX models; biology},
doi = {10.1115/1.1372322},
publisher = {Cambridge},
url = {http://link.aip.org/link/?PBY/321},
}
\end{filecontents*}
\documentclass[a4paper,12pt]{article}
\usepackage[american]{babel}
\usepackage{csquotes}
\usepackage[style=apa,
backend=biber,
natbib=true, % use this to use natbib cite commands
url=false,
doi=false]{biblatex}
\DeclareLanguageMapping{american}{american-apa}
\addbibresource{\jobname.bib}
\begin{document}
\citet{AD.Smith2001}.
\printbibliography
\end{document}

Non-recommended (but will work) solution
It's not that difficult to modify a .bst file to not print any of these fields. The .bst file plainnat has functions that look like this (one for URL, one for ISSN, one for DOI etc.)
FUNCTION {format.isbn}
{ isbn empty$
{ "" }
{ new.block "ISBN " isbn * }
if$
}
We can replace each one of these with a function like this:
FUNCTION {format.isbn}
{ "" }
And this will simply cause any of these fields not to be printed. Make a copy of plainnat.bst, give it a new name, (I've called it plainnat-noURL, and save it in the same folder as your .tex document. Then edit the file and for each of functions, replace that function with one like the one above.
With this minimal example we get the output we want.
\RequirePackage{filecontents}
\begin{filecontents*}{\jobname.bib}
@ARTICLE{AD.Smith2001,
author = {Arthur D. Smith},
title = {A simple model of LaTeX References.},
journal = {Journal of LaTeX},
year = {2001},
volume = {100},
pages = {1--10},
number = {3},
keywords = {LaTeX models; biology},
doi = {10.1115/1.1372322},
publisher = {Cambridge},
url = {http://link.aip.org/link/?PBY/321},
}
\end{filecontents*}
\documentclass[a4paper,12pt]{article}
\usepackage[round]{natbib}
\begin{document}
\cite{AD.Smith2001}.
\bibliographystyle{plainnat-noURL}
\bibliography{\jobname} % using the bib file created with filecontents
\end{document}

biblatex: fornatbibit will depend on which style you are using. What does your\bibliographystyleline say? – Joseph Wright Aug 26 '15 at 19:49natbibis a package for citation management and bibliography formatting. However, there is no bibliography style callednatbib. Moreover, there is no option calledroundedto pass to thenatbibpackage -- did you mean to writeroundinstead? Please edit your posting to clarify your setup. – Mico Aug 26 '15 at 20:42filecontentsto include the bib entrys to your MWE. – Mensch Aug 26 '15 at 21:10