I'm not seeing results you see:
Code:
$ cat in
"This is a test"
"Test1"
"Test2"
$ sed /\"/s/\"//g in
This is a test
Test1
Test2
Note that I replaced your useless use of
cat; nearly all utilities can read files specified as command line arguments.
Code:
$ sed 's/"//g' in
This is a test
Test1
Test2
Note that I removed the addressing portion (/\"/) as it is unnecessary in your case. No substitution could occur unless the line contains a double quote anyway, right? Also, it is more common to use single quotes to quote the entire sed expression, as it makes reading the expression clearer.