Given your comment that you want to replace an adhesive bandage, rather than medical tape:

Yes, you can use duct tape + CLEAN cloth for this, but you can do a few things to make it more pleasant.

1. Cut your fabric large enough so that the edges are far from the actual wound, this will reduce pulling on the wound when you remove the tape.
2. Only tape around the edges of the fabric. This lets the fabric stretch even if the tape doesn't (if your fabric isn't stretchy, cut it on the bias so that you get some stretch out of it).
3. Minimize the amount of tape that actually contacts skin. You only need enough to keep the bandage in place.
4. Even better, don't tape on skin at all if you don't have to. If you are bandaging a finger, wrist, or other similar area, you may be able to keep the bandage in place perfectly well by wrapping cloth entirely around and taping the bandage only to itself (still leave untaped fabric to provide stretch).
5. Test the stickiness of your tape on some non-injured section of skin. If it seems to be too sticky (painful to remove) then you can repeatedly stick/remove it from non-injured skin to reduce the stickiness before you put it anywhere near the injured area.
6. Flex your muscles while taping, this will make the tape wrinkly/slack when your muscles are relaxed. (The wrinkles are a bit annoying, but much more comfortable than the bandage being too tight.)

Note that this is probably not the most efficient improvised bandage that you can make. I would rather improvise with women's sanitary pads than cloth (these make really good improvised bandages and are usually individually wrapped) or glue (you should only use approved non-toxic glue on your skin, but really glue bandages are fantastic).

Given your comment that you want to replace an adhesive bandage, rather than medical tape:

Yes, you can use duct tape + CLEAN cloth for this, but you can do a few things to make it more pleasant.

1. Cut your fabric large enough so that the edges are far from the actual wound, this will reduce pulling on the wound when you remove the tape.
2. Only tape around the edges of the fabric. This lets the fabric stretch even if the tape doesn't (if your fabric isn't stretchy, cut it on the bias so that you get some stretch out of it).
3. Minimize the amount of tape that actually contacts skin. You only need enough to keep the bandage in place.
4. Even better, don't tape on skin at all if you don't have to. If you are bandaging a finger, wrist, or other similar area, you may be able to keep the bandage in place perfectly well by wrapping cloth entirely around and taping the bandage only to itself (still leave untaped fabric to provide stretch).
5. Test the stickiness of your tape on some non-injured section of skin. If it seems to be too sticky (painful to remove) then you can repeatedly stick/remove it from non-injured skin to reduce the stickiness before you put it anywhere near the injured area.
6. Flex your muscles while taping, this will make the tape wrinkly/slack when your muscles are relaxed. (The wrinkles are a bit annoying, but much more comfortable than the bandage being too tight.)

Note that this is probably not the most efficient improvised bandage that you can make. I would rather improvise with women's sanitary pads than cloth (these make really good improvised bandages and are usually individually wrapped) or glue (you should only use approved non-toxic glue on your skin, but really glue bandages are fantastic).

Re: What type of fabric/material to use... It's less dependent on the fiber content (cotton vs any other type) and more dependent on other properties of the material: clean (very important), not fuzzy/sticking to scabs too much but still being absorbent, being comfortable to the touch, moisture wicking/breathable, and being thin/light enough to tape down.

Natural fibers (cotton, linen, bamboo, wool, silk, etc.) probably get a slight edge up on moisture wicking/breathable, but how the material is created has more impact across the board. Something like a lightweight poly tee shirt would be better than a heavy cotton canvas. (Actually there are studies showing that spider silk is one of the best materials for bandages because it also has some antibacterial properties, but probably most people don't keep large quantities of clean spider silk laying around.)

If you dissect a commercial adhesive bandage you'll notice that they achieve all of this with multiple layers, one layer that prevents it from sticking to scabs and then an inside layer that does all the absorbing, and you can use that technique, too. Also note that I keep saying material instead of fabric. You could make a pretty good bandage with one layer of gauze backed up by a couple of layers of paper towel for absorption.