What effect would replacing the egg in a pie crust with egg whites have?

I would say that the effects will be too minimal to notice. So if you want to do it, go ahead and use up your egg whites.

Theoretically, egg yolks provide some emulsifying powers, and lead to softer, richer, sometimes even slimier doughs, while egg whites increase binding more and give you drier doughs. This can be quite noticeable in egg-rich cake types such as genoise or in yeast-leavened cakes. But in the case of a pie crust, it is almost completely made up from flour and butter, with the liquid needed only so you end up with a ball of dough with a bit of gluten, and not a bowl of crumbles.

The standard American crust uses water as the liquid in this type of dough, but when it is used for making shortbread cookies in European recipes, it frequently has an egg instead. I don't remember seeing eggs in tart crusts, maybe it is present in the Alsace. These doughs are not bad to work with, but with the small amount, and especially with flaky crusts deriving their texture from the butter pockets, I doubt that eaters will know that much of the difference. You certainly don't get a strong yolky mouthfeel when you use a whole egg, and with the crust being very fatty anyway, the missing yolk will be even less noticeable. So you are quite free in your choice of liquid, including egg white.

As a side note, if you are doing gluten-free crusts, you do need all the extra binding you can get, so there pure egg white is preferred over whole eggs or water.