There is a way to do this, but you may find yourself mired quite quickly.
The standard schemes have some "uncommitted" fields editora and editoratype and editorb etc which can be used for non-standard "roles". By default they get printed after the title, as "by ..." using a bibstring called byx where x is the relevant type, so if editoratype is redactor, it gets printed using the byredactor string.
To leverage this, in the standard styles, you need (a) to define a suitable by... string, because the ones you want are not pre-defined, and (b) to use it appropriately in the .bib file. It's easier to explain by example below:
\documentclass{article}
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@film{nemo,
editora = {Stanton, Andrew},
editoratype = {director},
editorb = {Walters, Graham},
editorbtype = {producer},
title = {Finding Nemo},
date = 2003,
}
\end{filecontents}
\usepackage[style=verbose]{biblatex}
\NewBibliographyString{bydirector, bydirectors, byproducer, byproducers}
\DefineBibliographyStrings{english}{%
bydirector = {Directed by},
bydirectors = {Directed by},
byproducer = {Produced by},
byproducers = {Produced by},
}
\addbibresource{\jobname.bib}
\begin{document}
\nocite{*}
\printbibliography
\end{document}
Produces

However, if you pursue this you will always certainly find yourself wanting to adapt the whole citation system to cope properly with films. (I was surprised, frankly, that @film even worked!) I haven't checked, but I have a feeling that the biblatex-chicago systems have something properly worked out for this. And there may well be other specialised systems, which if you want to use this a lot are probably worth pursuing.