If you are "resigned to do something", you agree or accept to do it even though you don't really want to. In a way, you see that thing as inevitable.
If you "put something at risk", you put it in a position where it may get harmed or damaged. If you go sky-diving you put your life at risk. If you rob a bank you put your freedom at risk (because you may go to jail).
Possible interpretation of "resigned to put her feelings at risk" (without further context or information): ...she decided to let herself enter an emotional arrangement in which she knew her feelings might get hurt