It's a tough question. I put "no gifts" on my first son's first birthday party invitation, but not because he had too much stuff - just because it was more of a grown-up party with our friends to celebrate surviving parenthood. But later on the same question came up in a different parenting forum, and now I think I wouldn't say "no gifts" because a lot of people seem to think it's impolite, and I guess I see the point - what happens is that some people do bring gifts, and that makes others feel uncomfortable. (It would be a lot worse for an older kid's party with friends, though.)
I think as a culture we are going to have to come to terms with the fact that we waste a lot of our resources on these toys that take up a lot of space, but I don't feel like the etiquette has caught up with the reality of our lives, where we want to celebrate with lots of friends and family but (in the kindest and most grateful spirit) don't need quite as much to store.
If people ask what you want, you can ask for consumable items like art supplies that you can always store and use for years to come. After the first day, I put away some of the gifts in the toy room closet for a "rainy day" or for a few months from now when they have forgotten about them. When my first son was born we got so many stuffed animals it was kind of ridiculous. I put away about 10 in a box, and they came out as "new" when I had son #2, who didn't have a shower so didn't get as many baby gifts.
Also, as a last thought, I can't tell you the number of favorite, wonderful toys my kids have received that I either didn't know about or would never have thought to buy myself.