If you have a web2c based system you can experiment with TEXMFOUTPUT
My copy of
% Write .log/.dvi/etc. files here, if the current directory is unwritable.
%TEXMFOUTPUT = /tmp
However it's probably not a good idea to do this.
.aux files for example need to be read back in by LaTeX so if you manage to move them you need to configure latex to find them in the non standard place. Similar considerations apply to other auxiliary files such as
.bbl, bookmark files written by
hyperref etc. Even
log files which are not (usually) read back by latex are often parsed by editors and GUI etc so that they can report and step through errors etc.