No you cannot get HIV from touching toilet handles or anything else that someone has touched.
It's not possible for HIV to be spread by touching toilet handles or other objects since HIV is sexually transmitted and besides you should always wash your hands after using the bathroom or touching the toilet handles anyway.
You can get other germs by touching toilet handles so I always use a piece of toilet paper to flush public toilets with instead of using my bare hands and I also wash my hands with warm water and soap to get rid of any germs.
Although HIV could get on a toilet handle or toilet seat the HIV dies pretty quickly once it's outside someone's body so the HIV cannot live long enough on surfaces such as toilet handles, toilet seats etc to infect someone else.
So you're safe using public toilets even if someone who has HIV has sat on the toilet seat.