Doing this for single line strings is very simple:
sed 's/((([^)]*)))//g' file
If you need it to deal with multiline strings, it gets more complex. One approach would be to use `tr` to replace all newlines with the null character (`\0`), make the substitution and translate back again:
tr '\
' '\0' < file | sed 's/((([^)]*)))//g' | tr '\0' '\
'
Alternatively, you could just use `perl`:
perl -0pe 's/\(\(\([^)]+\)\)\)//g;' file
The `-0` causes `perl` to read the entire file into memory (this might be a problem for huge files), the `-p` means "print each line" but because of the `-0`, the "line" is actually the entire file. The `s///` is the same idea as for `sed`.