It's important to understand exactly what's going on here. The ISO setting on a digital camera and on a film camera are kind of related, but they actually do quite different things. On a digital camera, broadly speaking, the ISO setting changes the sensitivity of the sensor to light. On a film camera, changing the ISO doesn't change your film in any way – it can't – ISO is a chemical property of the film. The ISO setting on a film camera simply tells the camera's meter what film you have inside, so it can give appropriate readings for that film speed.

So when you decide you want to push a film by 2 stops, what are you actually doing? You set the ISO to 400, which is actually setting the meter to ISO 400. But your film inside is still an ISO 100 film. The meter operates on the basis that you have a more sensitive film in the camera than you actually do, and so it selects exposure settings that are in fact too little for ISO 100 film. In effect, you are purposefully underexposing every frame by 2 stops*.

So what would happen if you didn't say anything to the lab about this? The film doesn't "know" you intend to push process it. It doesn't know it has been underexposed, and it has no way of automatically communicating this to the lab. So what you'd get is just a roll of underexposed photos.

So yes, you need to tell the lab that they should compensate for the film's underexposure by pushing the film 2 stops in development (i.e. your ISO 100 film was shot at EI 400).

* In general, I would not recommend doing this. Colour film especially just does not perform well when it's underexposed, even when you try to compensate through push processing. It's worth restating – don't be under any illusions – changing the ISO setting does not magically make your film more or less sensitive. Push processing is just a technique for using overdevelopment to get something better out of underexposure, at the cost of increased contrast, increased grain and potentially skewed colours. To get the best and most satisfying results, shoot (and develop) your film at box speed.