From what I understand, resolutions passed by the UN Security Council are legally binding on all member states compared to those passed by the General Assembly. So what would happen if a permanent member of the Security Council, like the US or Russia, were to disobey a resolution despite it being legally binding?
In theory, any number of things can happen: sanctions by the UN itself, sanctions by UN members, proceedings at an international court, etc. In practice: probably nothing. There is no rule of law in international law. If you are powerful, you are largely exempt from the rules. I'm not saying that to be political or cynical. It's hard to point to a single example of a great power facing serious repercussions for violating international law. I suppose the Soviets' invasion of Afghanistan led to sanctions, but that happened while the USSR was already declining; when the U.S. invaded Panama 10 years later, the worst consequences she suffered was a condemnation by the UNGA (the UNSC condemnation was vetoed by the U.S., France, and the UK—incidentally, the most recent time that the latter two powers used their veto).
Members of the security council don't have to ignore a resolution.
They can simply veto the resolution if they don't like it. All of the five permanent members of the security council (France, China, US, Russia, and UK) have veto power over any resolution, and that veto cannot be overridden.