The short answer is, well, not a lot.
While Arsene Wenger has criticised the new Premiership rules to be applied as of next season, Arsenal stands to benefit most as it has one of the best academy systems in the league.
The current Arsenal first team squad, with an average age of just under 24 years old, has 16 'home grown' players, double the minimum. With plenty of young talent coming through the system, this is unlikely to become an issue for Arsenal any time soon.
What is slightly more of a concern, if it is one at all, is the squad size limit of 25 players over 21 years old. As the squad currently stands, Arsenal will only have 21 players within that 25 limit for the 2010/11 season. This number becomes 25 for the season after, with Gibbs, Traore, Vela and Walcott moving into the limit. At this point, however, it is likely that Silvestre and Gallas (both 34 by then) will have moved on to be replaced by younger talent (e.g. Nordtveit and Bartley backing up Vermaelen and Djourou?).
All-in-all, the rules come at a fortunate time for Arsenal as the academy products under Wenger's system are just maturing into fringe first team players. While Arsenal's competitors will struggle with the new rules (particularly Liverpool and Chelsea, the latter compounded by the transfer ban), the Gunners will actually be able to add to their squad at a time when 'foreign players' will be less in demand and thus likely to be cheaper.
Arsenal's academy products who do not make the grade should also fetch higher prices as they become hot commodities.
Arsenal's principle of a sustainable club with a focus on youth development may have translated to a few bare years for the trophy cabinet as it competes against deep, unsustainable pockets. However, the Board and Wenger's patience should pay off finally as what many fans hope will become a 'golden age' for Arsenal over the next few years.
Here's to hope, at least.