What is the minimum Intel processor that won't bottleneck the graphics

Well this may be just me but i believe that it would be fine with the dual core but it really depends on the speed the Athalon is going at. i cant remember of the top of my head how fast it is in stock speed so you need to look into that but really you will bottleneck any GPU with a slow dual core thats why most people are with a x4

just so you know a Phenom II 940 is really cheep these days and works grate! almost as good as a core i7 when its OC!

Good advice up there^. Unless you pick up one of the old X2's running at least 3-3.2ghz, you should set your sites on the new Phenom Processors. They perform better clock for clock, and they run much, much cooler than the old X2's, and use less power.
I had a 4600X2, running at 2.8ghz. I am now running a Phenom II 920 quad at 3 ghz, and the difference, even in games, is substantial. I have never ever regretted retiring my old X2 rig, or the money I spent on my new Phenom II rig. Oh, and just so you know you can trust me, I am running a BFG 8800GTS 512 OC, which is "comparable" to your 9800GT. You should expect nearly the same results.

What is the minimum Intel processor that won't bottleneck a NVIDIA EVGA GeForce 9800 GT? I don't need the best processor from Intel, I just need one that won't slow down the graphics card.

I personally thinks anything above 2.6Ghz -Dual,triple or quad should be alright for you..I'm using the old 8800gs OCéd quite high & it should stay just below the levels of 8800gt=9800gt i should say, been using them starting from Athlon x2 5000 (2.6Ghz), 7750 (3.2ghz) and my current Phenom II 720 (3.7Ghz) for the past 6 months..In games, they do improve significantly especially loading times and reduced stuttering..

It's an 8400. Sorry, that's not a good quality graphics card. That's pretty much crap in fact.