The Fed may increase the rates. But will that make the government debt unsustainable? The US Fed is increasing its interest rate targets. These rates will eventually affect most other interest rates in the US economy. In particular, interest rates on US government bonds will also increase, which should increase the US government debt servicing costs. Moreover, the government has over 10 trillion dollars in debt, so, the effect on its costs could even force it to increase tax rates. So, is the interest rate increase by the Fed eventually going to lead to an increase in federal tax rates? Thanks!
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 !