No specific name is given to the "Heart Virus". When one is infected with the Heart Virus, symptoms include severe malaise and fatigue. Turning into a Super Saiyan also hastens the virus' detrimental effects. When Goku first felt the symptoms of the Heart Virus, he was out of breath just flying away from the island where the Androids appeared. As the infection progresses, the victim starts experiencing excruciating chest pains until eventually, they succumb to the virus and die. Senzu Beans will not stop this virus or even slow it down. The only way to cure this virus is with the Heart Medicine which was developed in Future Trunks' timeline at some point after the death of Future Goku. The Heart Virus is shown very potent as it is able to overpower even a Pure Blooded Saiyan's immune system (which is considerably stronger than an Earthling's) and will eventually kill them, provided they do not receive Heart Medicine in time to cure them of it. While never it was confirmed, Piccolo speculates that the virus might be contagious, and advises Yamcha to take some of the medicine himself after carrying Goku back to his house.
Future Trunks' timeline
Future Trunks mentions that many people succumbed from the virus in his alternate timeline. Unfortunately, Future Goku is one of the people who died from the Heart Virus and cannot be wished back to life with the Dragon Balls, due to his death being a natural cause. Six months following his death, the androids appear and murder all of the Z Fighters, with the exception of Future Gohan and the Z Fighters are unable to be wished back to life, due to the Dragon Balls being permanently unusable due to Future Piccolo's death, which also kills Future Kami, thus sparking the events of Dragon Ball Z: The History of Trunks.
In the main timeline, Future Trunks arrives from the future and gives Goku the antidote to the Heart Virus (the virus is incurable when Goku first gets it, but a cure is developed soon afterward; it was just too late for Goku) after telling him about the Android threat and that he will die of a heart virus prior the Androids' arrival. However, his presence in the past causes history as he knows it to change drastically. One major change is that Goku does not contract the heart virus when he was originally supposed to, but instead it appears several months later during his fight with Android 19 in May of Age 767. Goku is saved from death by Vegeta and taken home by Yamcha to receive the antidote. Ten days later, the virus is successfully cured, and Goku is finally recovered and re-joins the fight against the androids and the new threat, Cell.
It is never explained why or how Goku contracted the virus six months after his alternate timeline counterpart, though it was presumably one of the many side effects created by the time distortions from the use of Future Bulma's Time Machine by Future Trunks and/or Cell. It is unknown if the Heart Medicine Goku used to cure it was ever developed in the main timeline.
Video Games Appearances
In the Budokai series and Dragon Ball Z: Infinite World, it is called Viral Heart Disease and is a special capsule item that can only be used by equipping in the "custom" menu in the character selection screen. It takes 2-3 of the 7 technique slots, in game it causes both the user and opponent's health to deplete continuously, until only half of their last health bar is left. The user and opponent can avoid taking damage by equipping "Vaccine", a single slot capsule item which nullifies Viral Heart Disease's effect.