But would the modulating frequency of the LED confuse the Arduino logic?
Do you mean, would the 56KHz modulation of the LED in the XBox IR remote confuse the Arduino logic?
It depends on what the Arduino logic is.
Assuming the signal were fed into something like Timer1 input capture, or used in pulseIn, I don't think there would be a problem.
I did Google for "56khz IR Xbox remote" and got a bunch of plausible hits.
Also, what sort of issues would I be seeing if I was using a 40 kHz receiver with a 56 kHz receiver?
Well, that is a long way off, but maybe the Vishay datasheets I looked at were for their high-noise-immunity parts, so maybe 'normal' parts are more forgiving, and you were getting hunks of data? Maybe 'patches' within a pulse-train?
Having said that, I think the Vishay parts I have been using suggest only a few KHz deviation.
Sorry, I haven't got any IR remotes that far from 38KHz, which is what I've been using (and I don't have an XBox).