Wrong. Interest rates will rise if the Federal Reserve raises interest rates. They will not if it does not. The Fed controls 100% of overnight rates and virtually all of the rest of the yield curve[1]: "as we went out on the curve the correlations remained very tight (FFR at 100%, 3 month at 97%, 1 year at 96%, 5 year at 91%, 10 year at 86%, 30 year at 87%). Even at the longest duration there is an 87% correlation between the movements of the Fed Funds Rate and the 30 year bond. In other words, the bond market is the Fed’s whipping boy. Not the other way around."
What the Fed does not control is the US dollar's foreign exchange rate. So it's entirely possible that debt ceiling worries will cause a weaker dollar. Paradoxically though, they might cause a stronger dollar since the government not issuing debt actually constrains the dollar supply. Sovereign finances are deeply weird, and global reserve currencies are too.
[1] https://www.pragcap.com/i-want-to-come-back-as-the-federal-r...