Oops, I got confused. Actually Rossyd did not address the subject of the OP's question..... (unless I just missed it).
You did miss the answer. It was a simple question that allowed a simple answer.
Does aperture change induced focus shift vary with distance? I.e., can a lens exhibit focus shift at far distances but not close distances or vice versa?
Yes, it can change.
If you think you have an issue with auto focus accuracy and have a camera that allows you to fine tune the AF accuracy, do have a look at Reikan FoCal. It will perform tests on issues like this and the results are well presented for you to make an informed choice of what might be the best settings to choose.
As there's only a single value for AF adjustment on most cameras, you have to make a choice about what value to apply. If you have a set of test results, from FoCal or elsewhere, you can see where there might be an error and can choose an appropriate value for the type of photographs you routinely take.
For distant landscapes, you'd probably opt for a correction value at a long distance and small aperture, if you routinely shoot much closer and with wider apertures you would be better to choose a value for those settings.
If absolutely obsessive about it you could create a table of correction values for all settings to use.