What do you think?

I think its fantastic you want to share code ie contribute something :)

Do you think you'd like to use such a Perl module?

Nope, because it already exists :) ALMOST its called Capture::Tiny/IO::Capture::Stderr/IO::Capture::Stdout/Tie::STDERR/Tie::STDOUT

almost exists but not exactly identical to your usage/api

So is there room for your module on CPAN? yes there is room, CPAN not afraid of crowding :)

Do you like the name 'Auto::Log'?

No, it introduces a new namespace "Auto" and auto is a meaningless filler word, like "API" .. its a stopword ... yes, CPAN is full of APIs that do stuff automatically ... even log stuff "automatically" ... auto is one of http://pause.perl.org/pause/query?ACTION=pause_namingmodules#NAMES_TO_AVOID

Also, your module is not really a "Logging" module, all it does is redirect STDOUT/STDERR while most "logging" module provide granularity/filtering/timestamps...

You can learn lots about naming APIs by reading these nodes
http://perldoc.perl.org/perlnewmod.html#What-should-I-make-into-a-module%3f
Regrettable module names
Naming of modules that are mean to be inherited only?
Naming convention for variables, subroutines, modules, etc.. and variable declaration
Advice please on variable naming style in modules
Module naming problems, or time ill spent.
Module naming issues
Module Naming Conventions
Perl Naming Conventions
Module Naming Advice
Naming a Mail Action Module
CPAN Module Naming Suggestions.
Module naming advice
?node_id=3989;HIT=name%20mod;re=N
Module name suggestion
RFC: Best name for this module?
Namespace for local/internal modules?
Should I register a namespace in PAUSE before every cpan module upload?
Module's name and file's name
Names and ideas for new CPAN modules
Simplifying CPAN module namespace
Top Level Module Namespace Table
Hmmm, how should I name this new module/library ?
What's the general rule for naming modules with that extra 'x' in the namespace?
appropriate name/place for module which does JIT loading of modules
Registering CPAN module namespace
RFC: Name that Module
module name: Algorithm::RectanglesContainingDot?
RFC: Module namespace suggestion
Help with module name
Module Name Help!
Correct namespace of module I wrote
Namespaces and module suggestions again, OBDII
CamelCase in module names
What should I name my module?
Funniest Module Name
Changing a module's namespace on CPAN
Suggestion for module namespace assignment
What to name a module?
Re^2: Naming a module that handles SIP2 (library protocol stopwords)
Re: Submitting a module to cpan (Annovar::Wrapper Runner::Init )
Re: RFC: Name and/or API for module ("Type::FromData") ( infer sql best guess cast type match sqltranslator )
Re: Announcing Data::DynamicValidator
Re: cpan test redux...better, but... still shouldn't be on CPAN