Hi
I have a squeaky Whirlpool Ultimate Care 2 Heavy Duty Super Capacity Plus Dryer. After removing the rear panel I found that I can repeat the squeaky sound
By turning the lint exhaust fan.

Please tell me how I may fix my dryer myself.

Will I harm the dryer if I continue to use it until I am able to do the repairs?

Thank you for the help!

You will have to take it apart. You will have to clean the drier out thoroughly. Every air duct will have to be removed and scraped and washed out to remove baked on lint deposits which are like a concrete lining. I do not see the fan you mentioned in the parts list. It has a main motor which usually has sealed sleeve bearings, but if the felt pads at the ends of the motor have become dry there is an easy solution. There is a bell-shaped cover over the felt pads made of thin sheet metal. One trick I learned years ago was to take a sharp thin punch and punch about an 1/8” hole in the top of each cover, or use a drill/cordless screwdriver and drive a 5/8” head self-tapping screw in the top of each cover (since you will need to put a screw in each hole anyway). Make sure to blow away any metal shavings and vacuum them up. (If this is a ball bearing motor do not do this, replace the motor, but I kind of doubt that the problem is in the motor itself but if it's the main motor it could be in the belt tensioner roller or another roller.) Fill the felt pads with turbine oil, which you can get at any appliance parts or HVAC parts store for about $2.00 for a plastic bottle with spout, and while you are there get a tube of white lithium hi-temp grease. The hi-temp grease is for all other bearings in the dryer. This includes all support rollers and the belt tensioner. You may need to remove a c-clip on the end of the axles of the rollers to pull it out to grease it, but it's worth it to do so. I would also clean each axle and inspect it and replace if scored. I would also replace the belt while you have it apart.
It's very important to remove all air ducts that can be removed and to wash them out until clean of deposits. You will find these ducts where the filter is located on some dryers and sometimes up the rear where you will find a lot of build-up. Anywhere that air moves will need to be cleaned, some dryer doors even have a duct through which air is ducted. After reassembling you will now have a clean and quiet dryer.
Here are some manuals that may help: https://dl.dropboxusercontent.com/u/8010163/Whirlpool%20Dryers.zip

drtom4444 thank you for taking the time to offer your experienced advice on how I should repair my squeaky dryer.

You are welcome. If you have any other questions please feel free to post them. Good luck with your dryer and remember you need to clean this dryer about once a year to keep it running well.