Well, I am not sure there are performance improvements, just that Microsoft claims there are
Some folks have disputed the performance improvement claim by pointing out that even variables defined by var are still strongly typed, just not in the pre-compile sense.
A quick check of the Resharper settings indicates that you can change the notification for this.
Like I said, there is quite a variety of opinion on var, both for and against its use, much like any other tool that is in the programmers toolbox. You should consider the pros and cons for yourself.
For me, the pro is that as long as I am using self-documenting code, var can improve readability, and the biggest con is that the IDE does not always catch subtle differences in variable types and can lead to data loss or unexpected results, for example when implicit type conversion happens between ints, doubles, longs etc.
Also, keep in mind that you must initialize with a non-null value if you use var