The fancy term for this is cheilitis (it just means inflammation of the the lips). There are many causes for this. The most common is dry chapped lips but as you suggested, a superimposed infection (either bacterial or fungal) can occur. I'm assuming you aren't seeing any fluid filled bumps or blisters, right? I'm also assuming you haven't started any new medications/supplements, used new facial products (soaps, shaving creams,), etc? If you have, try stopping those for a week or so (if medically safe depending on the medication -- check with your doctor first in case they prefer you switch meds!) and see if that helps.
If not, I would follow these steps:
Lip balm or emollient for a week (Aquaphor works well). The trick to this is make sure your lips are completely blotted dry of any saliva or other liquid before applying. Otherwise you are just trapping moisture in there, which is breeding grounds for infection.
If that doesn't work, try a topical antifungal for a week (Lotrimin is ok).
If that doesn't work, try a topical antibacterial for a week (mupirocin works best, but may not be over the counter, so use Neosporin). If you use Neosporin and it starts to get worse fast, stop, because patients can develop an allergic reaction to Neosporin. Don't put on more Neosporin thinking you aren't using enough!
Lastly, try a topical over-the-counter steroid (hydrocortisone) for 1-2 weeks.
If none of that works, see the dermatologist. Also, don't worry. This isn't dangerous. Just annoying.
You can avoid this long drawn out process by seeing a dermatologist now who will scrape some of the dead skin off the surface of your lips and look for fungus right in the office. If they don't see any, they will recommend moisturizer and steroid cream right away and you should be good to go.
Hope that helps.
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