I have a GE Profile Model PSC23NGPA side by side refrigerator and recently the dispensed water tastes mildewed or something. I usually replace the filter and it corrects it. This time the filter didn't do the trick. Do I need to replace a water tank or something? If so, can I do it myself?

The bad taste in the water rarely comes from the refrigerator itself and 99% of the time comes from the water supply. If I had to guess, I would say that there is some kind of flexible line hooking your fridge up to the house. Either braided stainless steel or an opaque whitish rubber line and that will be the culprit. (attached pic shows both types)

They are usually used because they bend easily and are very abrasion resistant however I have found over the years that something within the rubber lining makes the water taste horrible! It does not happen right away and I have seen it occur between 8 months to several years after initial install date.

If you have this type then have it replaced with copper line only with precautions to avoid damage when moving the unit in or out. Install the filter bypass and flush it for several days by letting the icemaker run normally while dumping the bucket and getting glasses of water with at least a minute or two break in between so you do not overheat the water valve.

Let us know if this helps/corrects the issue

Get in contact with GE, they have a sanitizing filter you can install in order to run water into the sanitizer and that will run thru the water lines and hopefully clean out those line, also you can purchase sanitizer at Lowes and/or Home Depot and you can clean out the spout the water dispenses thru, PLEASE CAREFULLY READ ALL DIRECTIONS AND FOLLOW THEM