Nothing would happen to the new grass. The grass just wouldn't look as nice with the dead grass still in place.

The old grass would shade the new grass, slowing its growth. There's a bakery in our neighborhood where a bed was planted with switchgrass (Panicum virgatum) the year they opened. They never cut it back or did any weeding. The next year the old grass slowed down the emergence of the switchgrass and weeds pretty much took over. The same thing happened the next year. I'd say only about half of the switchgrass is still there.

Now it could be they died from some other reason -- lack of water or a weak cultivar, etc. To me it looked like they just got outcompeted, which I don't think would have happened (or if so, as much) if they'd been cut back.

It depends on the grass :-)) Deciduous grasses - those that look like straw in the winter - do benefit from being cut back annually if for no other reason than to remove the dead foliage and tidy up the plants. And the old foliage left in place could very well impede the development of the new growth. Evergreen grasses or grass-like plants - blue fescue or blue oat grass, various Carex species, NZ wind grass, etc. - do not need to be cut back and in fact, cutting back hard can limit their growth and in some cases even kill them. All these need is annual grooming or a raking or combing out of old foliage.

In my area people burn their grasses instead of cutting them. Of course you can only do that if they are not close to the house or other bushes.