Do Friday Night Magic games count toward your DCI rating?
Yes they do, but they have less impact on your rating, and a less serious level of rules enforcement, than tournaments at a higher level of play.

From the Wizards site:


  FNM events are 8k tournaments and have
  a Regular Rules Enforcement Level
  (REL).


Here's how that 8k figure factors into the calculation of your rating change after winning/losing a watch:

The Elo player-rating system compares players’ match records against
their opponents’ match records and determines the probability of the player
winning the matchup. This probability factor determines how many points a
players’ rating goes up or down based on the results of each match. When a
player defeats an opponent with a higher rating, the player’s rating goes up
more than if he or she defeated a player with a lower rating (since players
should defeat opponents who have lower ratings). All new players start out
with a base rating of 1600. The DCI uses the following equation to determine
a player’s win probability in each match:

Personal Calculation: 
                              1
Win Probability = --------------------------------------------------------------
              10^(( Opponent’s Rating–Player’s Rating)/400) + 1

This probability is then used to recalculate each player’s rating after the
match. In the equation below, players receive 1 point if they win the match, 0
if they lose, and 0.5 for a draw. Players’ new ratings are determined as
follows:

Player’s New Rating = Player’s Old Rating + (K-Value * (Scoring Points–Player’s
Win Probability))


For comparison, there are lots of more competitive tournaments at the 16k level, and Premium tournaments are at 32k!