Support the Old Joist Hanger
The first order of business is to support the joist, next to where you plan to remove the rusty hanger. You can place a board underneath the floor joist and towards the end where you are working and then nail the board to the adjacent floor joists. Also, it might be possible to support the floor joist from directly underneath, using cinder blocks, especially if you are working near ground level.
Remove The Old Joist Hanger
Once the joist is supported, then you can pull out the old rusty joist hanger. This is done not by tugging on the rusty metal, but by pulling the nails out one at a time, until the end of the joist comes free from the box frame that supports it. Pulling large framing nails out of an attached joist is not a simple task, so you have to be patient and pull each nail, a little bit at a time.
First, slide the curved end of a flat metal pry bar behind the nail and slowly pull the head of the nail towards you until the whole nail comes free. Repeat this process with each nail until the joist hanger comes free with no effort. Be aware that there might be a few short box nails that hold the flange of the joist hanger tight against the joist.
Put A New Joist Hanger In Place
When installing a new joist hanger, you will want to begin by setting the hanger in place and then with one hand placing a nail in the top hole of either side of the hanger. Make sure the top of the floor joist is flush with the rim joist or box frame. Also be sure that the nail is pointed at a 45 degree angle to the side of the floor joist and then start driving the other end of the nail with a 20-ounce framing hammer. Do this until the nail penetrates the floor joints and enters the rim joist (this is the long board that runs perpendicular to the floor joists).
After the top nail on one side is driven home, pull the joist hanger tight on the other side and start a nail through the top hole at a 45 degree angle to the floor joist. Drive this nail as far as it will go until the head is right up against the metal of the hanger. Complete the process for the rest of the nails (there will be six or eight total) that go through the joist hanger and penetrate the rim joist. Take down the support for the joist and you should be all set for that particular joist hanger.