Where (and why) do you need those escaped $ s? What do you want to achieve? There might be easier/better suited ways to get at it if we knew the entire picture.
For just your question, try
sed 's/\$/\\\$/' file
rm interfaces/DART/WEB-INF/classes/DART/util/TotalDisconnectUtil\$1.class
rm interfaces/DART/WEB-INF/classes/DART/util/TotalDisconnectUtil\$2.class
rm interfaces/RRIF/WEB-INF/classes/RRIF/util/Circuit\$Sect.class