I have a 2006 Raleigh Mojave 8.0 That I have ridden all over the mountains of Utah. I want to upgrade my bike but i'm not ready to drop $3000 yet. Right now i'm considering just putting a new front shock on my bike and "upgrading" to the equivalent Deore components that it has, but they happen to be in 10 speed instead of 9. Will there be an issue with trading out the rear cassette with the new 10 speed?

I've debated back and forth on the benefits of having a full suspension bike vs hard tail, and I haven't really let my bike stop me from going anywhere so i'm looking for the cheap upgrade on my current hard tail.

You'll need a new shifter to get any use of your new cassette. And a new chain.
IMO not great WRT cost/benefit.

The cassette will work with your existing wheels. You will need a new shifter and maybe derailleur. I think Shimano 10spd derailleurs use a different pull ratio than 9. But it may be road derailleurs that are like that.

You are looking at probably ~$200 to gain one gear. The biggest gain on 10spd was that they went to an 11-36 cassette. Do they make 11-36 nine speed cassettes? That may be a better option.

Yes they make 11-36 9 speed cassettes. Sunrace even makes an 11-40 9 speed cassette for $46 shipped from Amazon. The only real upgrade you get by going to 10 speed is the clutch RD. If you are going to bother upgrading you might as well skip over 10 speed all together and go to 11. There really isn't a price difference.

Not a lot of new options for 26er straight steerer forks out there anymore either.

In addition to parts mentioned, chainrings might need to be replaced/upgraded also. Start saving and don't put a lot of money into an 11 year old bike. Bike technology has changed drastically over those years.

There are lots of good options between $1000-2000.