I am replacing my chainset and chain, as its totaly worn and causing the chain to jump off when I pedal. The cassette is fairly worn too, but is not causing any problems. If I buy a new chain and chainset will these wear out a lot quicker if I leave the old cassette on?
You'll probably need to replace the cassette as well.
Generally once the chain has worn ("stretched") a little, it starts to wear down all the gears (front and back) that it touches. If there's enough wear that the front chainrings need to be replaced, it's almost certain that the gears on the cassette that you use most are also worn enough to need replacing. Usually the gears on the cassette will wear faster than the chainrings simply because they're smaller (so each individual tooth gets more wear).
The new chain will wear faster if the cassette is worn out, but a chain is generally cheaper to replace than a cassette. It should only wear the chainrings if you let the chain wear go.
You can look for signs of wear on the cassette, it will mostly be in the form of the teeth being thinner, often with a sort of "shark fin" appearance (sloped on one side). It's likely some gears (your favorites) will be worn more than others, which can give you something to compare against. If you can see signs of wear you should replace the rear cassette.
If you replace the chain and chainrings but not the cassette, I suggest paying very careful attention to how the chain runs when you first put it on. If there's any kind of roughness to how the bike pedals, any kind of noise coming from the rear end when you pedal, or problems with shifting the rear, then it's probably time to replace the cassette.
Another thing you can do if you keep the old cassette is check the chain for wear regularly. If the chain wears down ("stretches") faster than it seems like it should, replace the cassette.
First off, a good bike shop will have a tool for testing the wear on the cassette. Generally the rear sprockets wear out about twice as fast as the front, so the cassette is probably due for replacement if the front sprockets are gone.
After you replace the chain, you may notice that the chain and cassette "don't get along" very well together, with the chain tending to skip, jerking, not shifting well, etc.
Riding with the old cassette will likely cause the chain to wear faster, but not by much. Should not cause any untoward wear on the front sprockets unless you let the chain become too worn.