As those commands are defined in any of the \extra... commands of babel (if babel is being used) one needs to either apply such changes for all languages in use or (the better alternative) one patches varioref itself. Fortunately the commands are used only in a single place namely \@@vpageref. So a general solution would be:
\documentclass{article}
\usepackage{varioref}
\usepackage{etoolbox}
\makeatletter
\patchcmd{\@@vpageref} {\reftextfacebefore}{\unskip}{\typeout{*** SUCCESS ***}}{\typeout{*** FAIL ***}}
\patchcmd{\@@vpageref} {\reftextfaceafter}{\unskip}{\typeout{*** SUCCESS ***}}{\typeout{*** FAIL ***}}
% for \reftextafter and \reftextbefore we have to patch twice
\patchcmd{\@@vpageref} {\reftextbefore}{\unskip}{\typeout{*** SUCCESS ***}}{\typeout{*** FAIL ***}}
\patchcmd{\@@vpageref} {\reftextbefore}{\unskip}{\typeout{*** SUCCESS ***}}{\typeout{*** FAIL ***}}
\patchcmd{\@@vpageref} {\reftextafter}{\unskip}{\typeout{*** SUCCESS ***}}{\typeout{*** FAIL ***}}
\patchcmd{\@@vpageref} {\reftextafter}{\unskip}{\typeout{*** SUCCESS ***}}{\typeout{*** FAIL ***}}
\show\@@vpageref % have a look at the results
\makeatother
It is a bit crude as\@@vpageref now contains some absolutely useless code, ie a bunch of if-then-else statements without any code inside to execute. But who cares ;-)
The above also nicely works if you just want to disable the "face" ones but keep the ones where turning a page is required.
varioref + hyperref/nameref
If hyperref or nameref (a component of the hyperref bundle) is used in conjunction with varioref then the patching needs to be slightly different. In that case the macro that contains the calls is not \@@vpageref but \NR@vprageref. Furthermore it will get its definition only after \begin{document}, so the patch commands will look like this:
\makeatletter
\AtBeginDocument{%
\patchcmd{\NR@@vpageref} {\reftextfacebefore}{\unskip}{\typeout{*** SUCCESS ***}}{\typeout{*** FAIL ***}}%
\patchcmd{\NR@@vpageref} {\reftextfaceafter}{\unskip}{\typeout{*** SUCCESS ***}}{\typeout{*** FAIL ***}}%
\patchcmd{\NR@@vpageref} {\reftextbefore}{\unskip}{\typeout{*** SUCCESS ***}}{\typeout{*** FAIL ***}}%
\patchcmd{\NR@@vpageref} {\reftextbefore}{\unskip}{\typeout{*** SUCCESS ***}}{\typeout{*** FAIL ***}}%
\patchcmd{\NR@@vpageref} {\reftextafter}{\unskip}{\typeout{*** SUCCESS ***}}{\typeout{*** FAIL ***}}%
\patchcmd{\NR@@vpageref} {\reftextafter}{\unskip}{\typeout{*** SUCCESS ***}}{\typeout{*** FAIL ***}}}
\makeatother
Update/Correction
As Gabriel pointed out, has initial idea as well as my generalization of it, i.e., patching the commands \reftextafterand friends to simply do nothing has the side effect that you will end up with an extra space. The reason for this is that this space is generated automatically by \vref after the first number with the assumption that some text (such as "on the next page") will follow. So the correct way to deal with this is to patch the commands not by taking the mout but by replacing them with \unskip thereby removing that extra space again.
In the above code that is already corrected!