They're not explicitly immune to disease or poison, and they're not classified in any of the categories of creatures immune to disease or poison.
I don't remember if Celestials are immune, categorically, but if they are-- the greater portion of an aasimar is mortal flesh, and their celestial heritage doesn't provide any particular protection.
If you're not worried about editions, try checking out Warriors of Heaven (AD&D 2E) for more information about aasimars and other celestials. Likewise, if you don't mind third-party products, there's Fat Goblin Games's 5th Edition Racial Options - Aasimar!
Please note my use of affiliate links in this post.