Can you catch red's pikachu in HeartGold?

No, you cannot get Pikachu in Ilex Forest however you can find it in the Viridian Forest but there is a 5%-6% chance of encountering it, additionally you would have the option of trying to trade for it from Pokémon Diamond, Pearl or Platinum and you would be able to encounter it in the Pokéwalker through the Resort, Yellow Forest, Rally or Sightseeing courses and then transfer it over to HeartGold or SoulSilver however Yellow Forest, Rally and Sightseeing were event-only courses and Yellow Forest was the only event-only course that was distributed among the three of them.