Can I change my tyre size from a 20x2.20 to a 20x2.30 with out changing my rims
As long as you have wider rims it should be okay, but it really depends on the inner rim width. An inner width of 19 - 20mm will support a 2.30 tire just fine. If your rim has an inner width of 16 or 17 i wouldn't go any bigger, not that it would be catastrophic, but you increase your chances of a blowout on a big bump or curb. 

To find this measurement either look up the rims on google or measure it yourself, you're looking for the distance between the two bead hooks.

With that small of a jump though i imagine you will be just fine. Tube-wise you should be okay as well as they are made to handle a range of sizes rather than one specific size. One thing to consider is the more you pump a tube up the thinner it gets and the easier it will pinch flat and puncture. So without knowing what size tube you are running right now it's kind of hard to say. But many tubes are sold as a 20 x 2.00-2.40 so assuming these are what your running there should be no problem. If you have a 20 x 1.50-1.95 in there and it is already stretched past is usable range it may go boom. 

So with the assumption you have ample frame clearance (Check your brakes too) and that your are running the correct equipment to begin with you should have no problems.