Good notice! This can cause a series of questions with your home country. Not necessarily at the airport security, but once︀ you swap your expired passport for a new one. Quite a few residency programs involve︁ getting similar stamps. But they put it on the generic 'Visas' page, so you can︂ safely tear it out before you apply for a new passport.
If you don't want︃ any risk of getting questioned about it, tear out the page right after you get︄ your residency documents. The country, either UAE, or someone else, should not care about your︅ Visa stamp if you have a local ID card, but just in case, keep the︆ page separately in your wallet until you've re-entered the country at least once.