Use an alternate regex delimiter as sed allows you to use any delimiter (including control characters):
sed "s~$var~replace~g" $file
As mentioned above we can use control character as delimiter as well like:
sed "s^[$var^[replace^[g" file
Where ^[ is typed by pressing Ctrl-V-3 together.
Or else in bash shell you can use this sed with \03 as delimiter:
d=$'\03'
sed "s${d}$var${d}replace$d" file