Not sure if this is the best way of doing it, but here you go.
Code:
$ cat file
var="test string "is" string"
var2=test2 "is" also a string
$ sed 's/\([^=]*=\)"\(.*\)"$/\1\2/' file
var=test string "is" string
var2=test2 "is" also a string
edit:
awk and perl solutions:
Code:
$ awk 'BEGIN{FS=OFS="="} {gsub("\"$", "", $2);gsub("^\"", "", $2)};1' 'file'
$ perl -F'=' -lane '@F[1] =~ s/^"//;@F[1] =~ s/"$//;print join "=",@F' 'file'