I want to install a dishwasher in our 3br, 1ba (1400 sq ft) home, but our 100 amp breaker box is pretty tapped out. Can I add combine the dishwasher and garbage disposal on the same breaker (maybe using a tandem switch). Does the dishwasher need a dedicated line?

Also, at what point do we need to upgrade our service? We have mostly gas appliances (range, oven, hot water heater), but we do have a washer/dryer, a chest freezer, and central air (gas furnace). We recently finished our basement, adding 200 sq ft (from 1200 to 1400 or 1500), and I'm worried that we're pushing it.

100 amp service is the minimum residential service at this time.

To find out if you need to increase your service, you really need a licensed electrician to do a load calculation on your residence.

As to your question. Yes, both the D/W and Disposal can be serviced by the same branch circuit.

Also, if your D/W is on a cord it can be serviced by the same dual receptacle as the disposal, providing the amperage is correct.

I would suggest using a 20 Amp circuit carried on 12/2 AWG cable. A standard receptacle is fine. A GFCI receptacle is not required.

Can I do a load calculation myself, or at least a reasonable estimate?

Residential Load Calculations are fairly complicated and best left to the qualified. The calculations are
Described in Chapter 2, Article 220 of the 2008 NEC.

The reason I advocate using a licensed electrician is because the NEC is a safety manual. It is not a How To Manual. To make matters a little worse, most communities often have overlays to the NEC and you would have no way of knowing them.

Also, you need to start this conversation with your local Power Company. If they are not prepared to provide the upgraded service to you, you are dead in the water.

Laypeople cannot understand the National Electric Code, especially Article 220.

With all the gas appliances you have, I can tell you that you do not need a new service for a 1500 sq foot home.

You can safely add a dishwasher.

Odd that you have no breaker spaces. You can use a tandem breaker as long as the panel you have will allow it. Read the label on the door for the breakers your panel can accept.