Yes, the Lebanese War of 2006 followed the democratic peace theory, because Lebanon is not a very democratic nation. In order for the democratic peace theory to apply, the nations involved must be highly committed to democracy. Lebanon is not highly committed to democracy, and other things, like religious ideals, are more important to many in the country than democracy.
No, the 2006 Lebanon War did not follow the democratic peace theory, because the democratic peace theory says that democracies are not likely to go to war with each other. The theory says that democracies will choose peace, because they respect that the other nation has found democracy. When the Lebanon War happened, this was an aberration from that theory.