Does applying an OTA update remove rooted apps? I rooted my HTC One X just to get FM and other AT&amp;T blocked apps installed in /system/app directory. Now there is an OTA update and I'm sure it will unroot the phone. My question is will the FM app stay and work after the OTA? Thanks.
As eldarerathis already said in his comment, this very much depends on how the OTA is performed. First, some OTAs might even fail to execute, as rooting modified the /system partition and they check the CRC sum which then no longer matches (I had that with my HTC Wildfire/Buzz more than a year ago).

If they don't care about CRC, the answer still is "yes and no": Some OTAs are just "incremental" (replacing/adding single files); those wouldn't touch your root-apps, and root would be retained. However, full updates (and this includes most "version bumps", e.g. from Android 2.2 to 2.3 to 4.0) are completely "replacing" the /system partition; in this case root is gone along with all the other system apps.