Am I correct to assume that the tactile switches are a substitute for your IR sensors? If not what are they for?
I haven't used the ethernet shield but I wonder should you be creating a new instance of client in every iteration of the loop? I would be inclined to create the instance once in setup.
Am I correct to think that there is no problem when both switches are pressed provided one is pressed first and the problem only arises if both are pressed at the same time? Have you tried pressing them one after the other but with as short an interval as possible?
If your switches are each drawing 5v / 2200ohms = 2.3 mA then I can't see how they are overloading the 5v supply unless they are the last straw and the root of the problem is elsewhere.
Is it possible that when you press the two switches you move something else that has a poor connection?
It's an interesting problem.
...R