Government interest payments, which are already high, will become higher after future bond sales. This will compound future budgetary problems and could eventually lead to cuts in entitlements. If so, expect crime and political instability (already a problem) to rise in the future. This will take a while, though.
Normally rates are increased to lower inflation by reducing the supply of money. Given the multiple concurrent problems with energy (Hormuz, Red Sea/Yanbu, Russia/Ukraine, possibly Libya as problems are starting there, China is buying aggressively) then higher rates may not be enough to stop inflation. This would create a situation where both borrowing is harder and inflation continues to rage. This is very bad and will lead to demand destruction (nobody’s buying anything because it’s too expensive and they can’t finance it anyway). This results in a severe recession at the minimum.
Edit: wow, I really set off a discussion with this. See replies below for clarification on mortgage rates, which is the least important part anyways. Also, I should note that a lot of the above is a worst case scenario, if energy isn’t solved soon and especially if bonds don’t respond to the hike, leading to further hikes.