No, he counts nothing, as far as I can tell. Here's a quote from Mark Maske at the Post:
Redskins Avoid Cap Hit, Get Pick in Johnson Deal
The Redskins continue to use the restricted free agent market better than any other team, with today's trade of fullback Bryan Johnson to Chicago for a sixth-round pick in next month's draft.
The Redskins had until Friday to match an offer sheet that Johnson, a restricted free agent, signed with New Orleans. The Saints tried to dissuade the Redskins from matching the offer by making it salary cap-unfriendly. The deal contained an $800,000 roster bonus due Tuesday that, unlike a signing bonus, could not be prorated over the life of the four-year contract for salary-cap accounting purposes. It also included a temporary no-trade clause that expired Sunday. Because Johnson entered the NFL as an undrafted free agent, the Redskins would have received no draft-choice compensation for allowing him to go to New Orleans.
So the Redskins matched the offer sheet but then, between the period when the no-trade clause expired and the roster bonus became due, traded Johnson to the Bears to get something in return. Because the Saints' offer sheet contained no signing bonus, there was no salary cap hit for the Redskins to absorb in trading him. And they were left with three selections instead of two in the draft. . . .
