I can confirm from personal experience that travel to UAE, Syria, Jordan, Saudi Arabia is fine under tourist visa and during the stay. No one will ask about your HIV status, and it is best kept undisclosed. I was even able to rent apartments and open bank accounts (under applying laws for tourists of course). However, filing for residency or a work permit will require an HIV test, and that would result in deportation and travel ban. I can also confirm that carrying treatment/medication for 2 or 3 months worth is also fine, but not to exceed that amount or they will be questioned. Carrying them in their designated boxes is always safer than a random box or bag. I once was asked, what are these for and I said they are my immunity medications, and no further questions asked.
Restrictions only apply if you were tested in the country for a random test, a check up or residency. I, for example, have been in Jordan for 4 years, I leave every 3 months when my short stay allowance is done and come back in. As long as I am not tested, I am allowed.