How to use sed to remove all double quotes within a file

You just need to escape the quote in your first example:

$ sed 's/\"//g' file.txt

Are you sure you need to use sed? How about:

tr -d "\""

For replacing in place you can also do:

sed -i '' 's/\"//g' file.txt

or in Linux

sed -i 's/\"//g' file.txt

Additional comment. Yes this works:

    sed 's/\"//g' infile.txt  > outfile.txt

(however with batch gnu sed, will just print to screen)

In batch scripting (GNU SED), this was needed:

    sed 's/\x22//g' infile.txt  > outfile.txt