Actually, I have never seen the commando get damage from blowing up a building. What happens is minigunners spawn and shoot at him. You may not notice if its just 1 minigunner because the commando will fire back when attacked. But if 2 show up then it is obvious. This is what they changed in future games. No infantry spawn with C4 in Red Alert and that is why Tanya doesn't get hurt.

@Mikeagan indeed you're correct, the problem in Tiberian Dawn is that after a Commando unit blows up any structure with C4, one or more enemy infantry units spawn where the building used to be. These newly spawned infantry then proceed to shoot the player's Commando unit with their rifles. It's extremely frustrating in Commando missions since these spawned infantry units can reduce the player's Commando's heath points significantly (or kill the Commando) before the player has time to react and order the Commando to kill the spawned infantry. Hopefully cfehunter will either remove all infantry spawning after a building has been blown up by C4, or change something else to prevent this issue from happening in a future build of this excellent mod. :)

In the meantime the spawn seems to be somewhat random so you can save scum by quicksaving just before you blow up a building and reloading until no enemy spawns.