The lender might pay the taxes. They usually have some money in escrow for that purpose. If the taxes are not paid in Illinois they will be sold. The lender will receive notice of the sale and might pay them even if the escrow account has been depleted.
If they are sold, the party who buys them must wait 2 years to see if they are redeemed. If not, that party can go to court and ask to be given title to the property. The mortgage lien may still be in place at that point but the previous owner will no longer have title to the property. As you can see this takes several years to play out.