If I can't find the hole causing a flat tire, do I need to replace the tube?

To answer the question directly, yes, if your tube is losing air that quickly, it needs repair. It is not a matter of simply being too old.

There is likely a very small hole or a leak in the valve.

Replace the tube, or patch it. Benzo's soapy water suggestion will work very well.

Take the tube out and pump it up so that it balloons to 2-3 times its normal size. There have been times when the puncture is too small to detect in normal situations, even soapy water, but this has never failed me! Even the smallest puncture, like from a brake cable sliver, will show itself. Listen for the hiss and then mark it with a pen to find it after you deflate it (Usually I make 4 lines coming out away from the hole, like an X but with the middle taken out, this pic is just an X, but the 4 lines makes it easier to pinpoint).

If this doesn't produce any results it might be in the valve. If you have a Schrader, get your finger wet and put it lightly over the top (I spit on the ground and put my finger in it) this will allow you to feel any air leaking out. With a Presta it is a little harder, but wrap your wet thumb and index finder around the top of the valve to check it. Most of the time you can fix the Schrader using a core tool, not sure you can do anything to fix a Presta (but aside from a bent core bolt I've never seen a Presta valve leak)

My favorite method is to remove the tube from the tire, reinflate it a bit, and completely submerse it in a swimming pool. I've used this to quickly identify leaks that were so slow they took days to deflate the tube.

In my experience (twice) : To find where the puncture is based I get a bucket of water and have the inner tube inflated to the point where i can bend it into the bucket sections at a time. Naturally I see bubbles on the tube under the water at the bends in the tube due to trapped air... but if you squeeze the tube you should find bubbles rising. Move the tube out and put in a new section, repeating around the entire tube to make sure there isn't more than one puncture. Around one bubble per tire squeeze should be apparent when you have found the issue.

Before replacing the tire, you could try putting slime tire repair sealant in the tube. In my experience, slime is very efficient in closing the hard to find punctures from the inside.

After inserting the slime, pump the tire to the max and ride your bike for a while and see if it works.