SET ANSI_NULLS - Performance of Stored Procedures
This does not affect the performance of stored procedures directly, but in general it can affect performance indirectly, since SET settings used at execution time can influence the plan that is chosen. Two different users, one with the setting on, and one with the setting off, will get their own copies of the plans, and they may not even be the same plan (depending on other variables at runtime, like parameter values). See Erland Sommarskog's post, Slow in the Application, Fast in SSMS? On top of differently performing queries, this can lead to potentially different performance and will create two different plans in the cache, wasting memory. 

Eventually, you won't even have the ability to set ANSI_NULLS to OFF; see the documentation:


  In a future version of SQL Server, ANSI_NULLS will always be ON and any applications that explicitly set the option to OFF will generate an error. Avoid using this feature in new development work, and plan to modify applications that currently use this feature.


So, you should already be in the habit of always setting it to ON. If it's just always on, then it won't be an issue.