If the consensus on Jenny Curran's death in Forrest Gump is HIV or Hep C, wouldn't both Forrest and little Forrest have it also? It doesn't seem feasible to me that she could have had either one of these viruses without passing it to her son, and possibly Forrest.
Without treatment, the chance of passing HIV on to a baby is 25% (source). Seems absurdly low to me but I am not a doctor. As far as I remember, Jenny and Forrest senior only had one sexual encounter, so it is possible he didn't contract the virus either.
We are not explicitely told, so it's up to the spectators. In my mind I let the Forrests live a happy, disease free life.