How do I request a file but not save it with Wget?

You can use -O- (uppercase o) to redirect content to the stdout (standard output) or to a file (even special files like /dev/null /dev/stderr /dev/stdout )

wget -O- http://yourdomain.com

Or:

wget -O- http://yourdomain.com > /dev/null

Or: (same result as last command)

wget -O/dev/null http://yourdomain.com

Curl does that by default without any parameters or flags, I would use it for your purposes:

curl $url > /dev/null 2>&1

Curl is more about streams and wget is more about copying sites based on this comparison.

Use q flag for quiet mode, and tell wget to output to stdout with O- (uppercase o) and redirect to /dev/null to discard the output:

wget -qO- $url &> /dev/null

> redirects application output (to a file). if > is preceded by ampersand, shell redirects all outputs (error and normal) to the file right of >. If you don't specify ampersand, then only normal output is redirected.

./app &>  file # redirect error and standard output to file
./app >   file # redirect standard output to file
./app 2>  file # redirect error output to file

if file is /dev/null then all is discarded.

This works as well, and simpler:

wget -O/dev/null -q $url