How do I change a John Deere 155c's rear tire?

Hi,
I gather you mean an LA155C. It changes the same as any. Remove the wheel/tire from axle; don't lose that key that holds them together!
Remove the core from the valve stem, break the beads loose on both sides, lay flat on ground, push tire on one side to ride into the recess at the center of the wheel and pry the tire up, over and off on the other side and work your way around to free one side. Lift tire up so remaining side is against side of rim where tire was just removed and repeat.
Lubing the bead of the new tire will aid in installation, and reverse the process.
Not for the "faint of heart" or uninitiated! It is well worth it if a mower or tire dealer with a machine will do it for you for $8 or so. It takes a skilled mechanic with the proper machine less than 5 minutes. Expect to buy a new valve stem and do not expect that they will necessarily get someone to just drop what they are doing, especially if you show up in the middle of the day.
Peace,
Clarke

What I would love to do is to get the tire off so that an expert can take it and repair some holes in it. But it had a yellow cap that we pulled off and now it has something that we need to take off it looks round in the middle with a washer.
This is a 155 ca. It appears that the valve has popped off. The washer doesn't want to come off though because of something holding it on but I cannot seem to pop it off.

Hi,
Should be an E-clip. If you look at it, it should be shaped like a rounded "E" and have a couple of places in the areas between the ends and center where there is a slot of sorts. A thin bladed stright screwdriver should slip in and twist to ease it outward.
Peace,
Clarke

I got that off and the washer now if I jack up th mower I should be able to pop off the tire and the wheel well to the people to fix it? It just doesn't fit into the back of the pick up to bring it in that is why we need to get it off. I really do appreciate your help!! Thank you.

Hi
As I said, do not lose track of that key 5/32" square by 2" long. When you put it on, slip wheel and tire on, line up slot in wheel with slot in axle and slide in that key. Then your washer(s) and clip.
Peace,
Clarke