Use this sed command to achieve that:
sed '/^#/,/^\$/{/^#/!{/^\$/!d}}' file.txt
Mac users (to prevent extra characters at the end of d command error) need to add semicolons before the closing brackets
sed '/^#/,/^\$/{/^#/!{/^\$/!d;};}' file.txt
OUTPUT
# ID 1
$ description 1
blah blah
# ID 2
$ description 2
blah blah
blah blah
Explanation:
/^#/,/^\$/will match all the text between lines starting with#to lines starting with$.^is used for start of line character.$is a special character so needs to be escaped./^#/!means do following if start of line is not#/^$/!means do following if start of line is not$dmeans delete
So overall it is first matching all the lines from ^# to ^\$ then from those matched lines finding lines that don’t match ^# and don’t match ^\$ and deleting them using d.