Support got back to me with a solution:
Bascially, DreamHost’s servers don’t natively support HTML server side includes for .htm or .html files. However, there is a fix.
You can either rename the files to use the .shtml extension, which DOES support SSI…
or you can add an .htaccess file with this line – choose the version that matches your HTML file extensions:
AddHandler server-parsed .htm
AddHandler server-parsed .html