Could a hike in interest rates cause a country to be unable to pay back it's debt burden? Sometimes I read in news articles, that a central bank can't hike the interest rates much higher, because then the government would be unable to pay the interest rate burden of it's government debt. 
Is this really true? In my understanding this should only be true for government debt that will be issued after the interest rates are already high.

I mean, the biggest part of government debt that has been already issued is only traded at the secondary markt at a fixed nominal interest rate.
(except of an usually much smaller part of floating rate notes)
The short answer is no, rate hikes have no effect whatsoever on whether a government can service its debt and repay maturing issues.  Here's why---


A government that issues debt in its own currency can always print currency to make its debt payments.  This is an extremely important point, despite that certain political actors would like you to forget it.
Central banks set rates at the short end of the Treasury curve.  They do not have an effect on the long end of the curve, which is set by market participants' expectations of long term inflation and real growth.  What's more, central banks today are data dependent---they state explicitly that they will react to data that the market already has priced.  For example, this month the U.S. Fed's FOMC will likely increase rates due to stronger economic data (GDP revisions upward, better than expected inflation, decline in unemployment and jobless claims). However the Treasury curve has already steepened massively over the last month in response to these data releases.
Many governments issue debt in foreign currencies like USD and EUR (e.g., Brazil, Mexico, Ukraine, and almost every other emerging market).  Because of uncovered interest rate parity, rate rises push up the value of the domestic currency relative to the currency in which external debt must be paid.  This actually decreases the cost of servicing the external debt.