If the Blu-Ray burner is a multi, and can also burn DVD's then this is possible, but as Rohan23 says, a blu-ray disc will NOT play on a DVD player of any type.
As for your question, it is a tad bit confusing.
No, you do not need your monitor/video card/CPU upgrading if you are just going to burn Blu-ray discs, as this would be just as physically intensive on your CPU as burning a regular DVD, and won't affect what monitor/graphics card you have.
If on the other hand, you are going to be editing/converting HD content using Nero to then burn onto disc, and aren't JUST physially burning the disc, then you will need a better CPU than if you were not, as more pixels being rendered/edited will push the system more.
I think in truth, you yourself is a bit confused as to what you'd need to have anyway.
A DVI port is a digital signal and will in no way affect the burning/editing/converting of Blu-Ray discs.
DVI is just like HDMI, but does not carry sound along with it.
DVI is capable of sending higher than HD resolution content across to the monitor for use in games, etc.
I do not even think, unless the HDMI 1.1 standard is carried through to PC drives as well, that having a DVI monitor would stop you watching HD content from a blu-ray disc this way either.
You are right in thinking though, also, that if playing HD content from a blu-ray disc using your PC, a better CPU/graphics card would be needed than your average afair.
The 9600GT isn't that bad of a card (and is more than capable of producing a 1080p image), and as long as you had a good processor to boot, probably would not run into problems, as long as the HDMI 1.1 standard does not affect PC drives (which I'm not 100% sure of, but shouldn't). If a PC Blu-Ray drive did need the HDMI 1.1 compliance, then you wouldn't get a picture at all while playing back HD content using a blu-ray drive, but I wouldn't think for a moment, considering that a PC can record from within itself anyway, that this protection would be used.
If playing using your PC, your monitor would also be best if it could at least play 720p content, which ALL can, as HD resolutions have been used in PC monitors for AGES, and TV's have only just started using the technology.
1080p native monitors on the other hand, in the PC sector I believe are new. Yes, there are 16:10 monitors (16:9 is native in TV terms for producing the correct scale), and also PC monitors have been seen to go way past the 1080p needed for HD content.
But as long as your monitor can at least go up to 1280x720 (or similar) then you can play 720P content fine (all would be capable of this, but better if producing a widescreen 16:9 ratio). If though you wanted play at full 1080p, true to content, and so in scale, you'd need a monitor with a true 1080p (1920x1080) resolution, and not a 16:10 monitor (which will add more black bars, than a 16:9 monitor would). These are becoming more available if you ever decided to do this.
I hope this helps.
