Not really sure what you are looking for here.
You need a form where the user will set the filtering criteria. As John suggest above, you might put this form in a dialog, or maybe a popup or just on the same page with the list.
You need some JS that will set change callbacks on all of your filter criteria. The change callbacks can all call a common function that will then filter the list based on your criteria. Iterate through your list, examine each entry, and apply your filter criteria. If a particular <li> is to be filtered out, then set a CSS class that will hide it.
It might be useful to add some hidden data to your list items, to make it easier to access the filterable values. I'd suggest using data- attributes.
You will have to write the code to do this. Were you expecting us to do it for you?