Tip burn in dracaenas can be caused by a number of factors. I'll list them and maybe you can decide which might be the scenario for you.

Most dracaenas (yours is Dracaena sanderiana) are sensitive to fluoride in tap water. This can be made even worse by allowing the plant to dry out too much in between watering, or keeping the plant too moist. Tip burn can also be caused by growing a plant in a container that does not drain freely, allowing a build up of soluble salts in the potting medium. Over fertilization can be a causal factor, too. Improper watering can sometimes cause problems....by giving a plant little sips instead of a good, cleansing drench.

Rhizo: The pot the Dracaena is in does not have drain holes, but the Azalea that is having the same symptoms, does. The potting soil has quite a bit of perlite, and the pot is pretty deep, so I don't think the roots are sitting in water. We moved here in late March and the water/ground here is very alkaline, so I'm thinking that maybe the high salt content could be the problem. Should I buy one of the Brita water pitchers that filters out the minerals to water my plants with? I haven't fertilized in over a month, and am afraid to, until I solve this problem. Don't want to overstress a sick plant. I have been trying to water more thoroughly and less frequently, instead of giving the plants little sips. One of the many things I've learned from this wonderful forum : )

Hmmmm. Perlite can also be a source of additional fluoride. Eeek. Brita filters do not remove fluoride from your water.

The problem with containers without drainage holes is that you cannot occasionally flush (leach) the pot. That's the only way to remove excess soluble salts and to get fresh oxygenated water through the soil column. Some plants can adapt to this (as I can attest to), but others are more sensitive. I suggest that this is one of the problems with your dracaena. The fluoride is the other problem.

So I should repot it then. Maybe I could use a plastic pot with drain holes inside this pot, since it's one of my favorite pots. What do I do about the water situation?

I like the pot, too. ;-) Your plant might even appreciate being in a slightly smaller inner pot, so it will fit easily into the pretty one.

Don't add extra perlite to your potting mix, though I know that it's difficult to buy a good medium without it. Use bark fines for extra aeration, if needed. You can also mix up a tiny bit of lime into the potting medium before potting up the plant.