OK first of all and easiest you can do it with Sniper lvl 7. His range is bigger than the fountain's range and you could easily do it as long as nobody is bothering you. There is no other ranged hero who can destroy it solo.
For melee heroes, the easiest way is to buy Vladimir's offering as it grants lifesteal from the fountain.
Leoric is best suited for this as your passive lifesteal works on the fountain so you won't even need Vladimir's — just damage. Though Vladimir's makes it very easy.
Rikimaru is probably in top three due to the 75% miss from smoke. Just get butterfly and some rapiers and you should do it. Attack until 30% then smoke: you will be almost full. Lather rinse repeat.
Last one which worked for me is Faceless Void, although he is a little harder as Chronosfere doesn't work on fountain. Passive skill for damage block and a butterfly, full rapier should do it.
This heroes worked for me, but I've also seen people succesfuly soloing it with heroes like Ursa or Syllabear(best full item build hero 12 item slots).
If you are not that selfish, bring on a Crystal Maiden with you. She can cube the fountain making it easy to kill with any hero. Another good support to kill the fountain would be Omniknight due to ultimate and heal.
Axe's Culling Blade is the only ability that kills through Shallow Grave. Techies is also capable of killing themselves when under the effect of Shallow Grave by using Suicide. The Shallow Grave ability is also not able to be purged, so there is zero worry other than that.
Some spells will poison/affect you for longer than the 5 seconds that Shallow Grave lasts and thus may kill you as the effect ends if you're not careful.
Best Answer
BKB, Omniknight's Repel, Lifestealer's Rage, Juggernaught's Blade Fury, and the Earth Element of Brewmaster's ulti.
Edit: The first being obtainable on any Hero, the rest being situational as 'counter picks' against her.