I don't think you can really tell people not to bring gifts. It just makes it uncomfortable for everyone. People who don't bring a gift feel guilty, those who do feel guilty, and anyone who really doesn't want to give one won't bother (or very rarely, won't come). You can't specify what you want/need either (I know wedding gift registries have taken this to an uncomfortable level, and it's even worse when people ask for cash toward their honeymoon and not a lasting gift). You just have to be gracious.
So yes, absolutely have a house warming party! You are excited about your home, you invite people in to see and share your joy. Period. Some people will bring a gift, but it's not really different from a standard hostess/host gift - some will bring a bottle of wine, some may bring flowers (be sure you unpack a few vases just in case so you aren't scrambling!), some will bring a card, others may or may not bring a household item. No, you aren't setting up a household for the first time (but then again, neither are most newlyweds anymore!), so you don't need a toaster and a vacuum - but you won't get those anyway.
Just because you can afford to buy a blender doesn't mean you say to people "I have money so I don't need your charity" - which is how many will take the "no gifts" request. Some people will say something like "your presence is present enough" but even that is getting overdone. It's not really proper etiquette even though people do it so much.
Your true friends will know that this is not a gift grab - it's just a wonderful occasion to celebrate. Let them share in your joy and excitement in any way they wish. Some will joke that you probably spent your last time on your mortgage and therefore need a few things, others will just express their happiness with a little something. Accept it!
If you want to make it seem a little more casual, you can call it an "open house" so people know to drop in whenever they want, but then you don't usually ask for RSVPs. That can make the food prep a little iffy because you have no idea who's coming and when.
Honestly, I would invite the people I know and love the best, assume that they will be just as thrilled for you as you are for yourselves, and let them come. When you have birthday parties, do you specify "no gifts"?? I just don't think there's a nice way to tell people what to do - you let them do what they feel is appropriate, and then you thank them. Go ahead, enjoy it!!