Non-trailing newlines are not removed
The newlines you are looking for are there, you just don't see them, because you use echo without quoting the variable. 
Validation:
$ a=$( df -H )
$ echo $a
Filesystem Size Used Avail Use% Mounted on /dev/sda3 276G 50G 213G 19% / udev 2.1G 4.1k 2.1G 1% /dev tmpfs 832M 820k 832M 1% /run none 5.3M 0 5.3M 0% /run/lock none 2.1G 320k 2.1G 1% /run/shm
$ echo "$a"
Filesystem      Size  Used Avail Use% Mounted on
/dev/sda3       276G   50G  213G  19% /
udev            2.1G  4.1k  2.1G   1% /dev
tmpfs           832M  820k  832M   1% /run
none            5.3M     0  5.3M   0% /run/lock
none            2.1G  320k  2.1G   1% /run/shm
$ 
Trailing newlines are removed
As @user4815162342 correctly pointed out, although newlines within the output are not removed, trailing newlines are removed with command substitution. See experiment below:
$ a=$'test\n\n'
$ echo "$a"
test
$ b=$(echo "$a")
$ echo "$b"
test
$
In most cases this does not matter, because echo will add the removed newline (unless it is invoked with the -n option), but there are some edge cases where there are more that one trailing newlines in the output of a program, and they are significant for some reason. 
Workarounds
1. Add dummy character
In these case, as @Scrutinizer mentioned, you can use the following workaround:
$ a=$(printf 'test\n\n'; printf x); a=${a%x}
$ echo "$a"
test
$ 
Explanation: Character x is added to the output (using printf x), after the newlines. Since the newlines are not trailing any more, they are not removed by the command substitution. The next step is to remove the x we added, using the % operator in ${a%x}. Now we have the original output, with all newlines present!!!
2. Read using process substitution
Instead of using command substitution to assign the output of a program to variable, we can instead use process substitution  to feed the output of the program to the read built-in command (credit to @ormaaj). Process substitution preserves all newlines. Reading the output to a variable is a bit tricky, but you can do it like this:
$ IFS= read -rd '' var < <( printf 'test\n\n' ) 
$ echo "$var"
test
$ 
Explanation:
- We set the internal field separator for the read command to null, with IFS=. Otherwisereadwould not assign the entire output tovar, but only the first token.
- We invoke readwith options-rd ''. Theris for preventing the backslash to act as a special character, and withd ''set the delimiter to nothing, so that read reads the entire output, instead of just the first line.
3. Read from a pipe
Instead of using command or process substitution to assign the output of a program to variable, we can instead pipe the output of the program to the read command (credit to @ormaaj). Piping also preserves all newlines. Note however, that this time we set the lastpipe shell optional behavior, using the shopt builtin. This is required, so that the read command is executed in the current shell environment. Otherwise, the variable will be assigned in a subshell, and it will not be accessible from the rest of the script.
$ cat test.sh 
#!/bin/bash
shopt -s lastpipe
printf "test\n\n" | IFS= read -rd '' var
echo "$var"
$ ./test.sh 
test
$