A logo I made in Illustrator does not distort when you change size, but it is desirable to check whether the details remain readable in all various sizes. Are there any standard sizes that I should check the logo? What file size should be opened in order to do these tests?
As Scott says, there are no standards...
Testing at a pretty full range of sizes isn't much work though... In Illustrator (or any image viewer) it takes a few seconds to zoom on your logo from 3% to 6400% (or whatever the minimum and maximum zoom levels are). That'll give you a good idea of how it works. Then do some test prints at a few sizes from very small (a few cm max) to a full sheet of paper* (whatever size paper you have available to you). Unless you have any specific size requirements for the job then that'll do you pretty well.
* There's no real need to test much bigger than that unless you know the logo will be seen at a unusually large scale close up. Even if your logo is printed on a 20 foot billboard it'll usually be seen from a distance.