According to the Google Shell Style Guide, I should:
Always quote strings containing variables, command substitutions, spaces or shell meta characters
$(echo foo bar) is indeed a command substitution. In this specific example, you don't need double quotes because a variable assignment creates a “double quote context” for its right-hand side, so VAR=$(…) is equivalent to VAR="$(…)".
In bash, you don't need double quotes in export VAR=$(…) or declare VAR=$(…). But you do need the double quotes in some other sh implementations such as dash.
You do need double quotes in env VAR=$(…) somecommand, in make VAR=$(…), etc. It isn't the equal sign that makes the double quotes optional, it's the fact that the equal sign is parsed by the shell as an assignment.
There are a few other contexts where the double quotes are optional, but you can't go wrong with the simple rule: always use double quotes around variable and command substitutions unless you want the split+glob operator.