The Fed may increase the rates. But will that make the government debt unsustainable?
You have to differenciate the debt that is already existing from the one that is going to be issued: whenever the US Treasury issue a bond lets say a 10% coupon 10yrs... for a 100USD. It's still going to pay 10USD to the holders during the 10yrs. When you see the yield going up or down it's usually on the secondary market. I say usually because sometimes the government issues bond with a floating rate which can be impacted by a rate increase.

As for the new debt that is going to be issued you are right to say that this going to be more costly !