Illinois isn't likely to file a suit against you in Washington. If they decide to go to court to get a judgment, they'll likely do it in Illinois and serve you with notice of the suit in Washington. Then you have the choice of responding to the suit in Illinois (which would require hiring an Illinois lawyer to represent you there) or ignoring it and letting it go to default judgment. If the state of Illinois gets a judgment against you in an Illinois court, they can then come to Washington and file a foreign (out of state) judgment against you. That would give them the same power to enforce the judgment that they would have if they had obtained it in Washington state.

Whether Illinois has jurisdiction over you as a Washington State resident with a delinquent Illinois tax bill is a matter of Illinois state law, to be discussed with an Illinois attorney.

Not sure why IL would sue you in WA when they can just have WA enforce a proper IL judgment. You state they don't have a judgment. Are you sure? if they have not proceeded to get a judgment maybe they are not that interested in collecting.