A lot missing here. what is worst case dewpoint and temperature of the space and the air in the duct? Pick the most humid season.
If you are in the rainforest, and the duct provides 40°F air, it will be tough. If you are in Saudi Arabia and the duct just provides outdoor air and that matches the interior space, it will be easy.

The details are missing on purpose because as mentioned it's a conceptual question. Intuitively it's easy to understand you'll have condensation in rainforest but not in the desert. I'm looking for a generalized methodology and equations to evaluate the problem and calculate the required thickness in any application.

To make it more concrete, let's consider two examples:

1. Room in cold and dry climate

• Outdoor air: -40 C, 100% RH
• Indoor air: 18 C, 50% RH
• Duct: 500mm x 500mm, 5 m long, 200 L/s, -40 C supply air

2. Room in warm and wet climate

• Outdoor air: 30 C, 100% RH
• Indoor air: 18 C, 50% RH
• Duct: 500mm x 500mm, 5 m long, 200 L/s, 30 C supply air

What are the insulation requirements for each situation to prevent condensation on both inside and outside of the duct? Would you use external or internal insulation in each case?

I never would use duct liners since they deteriorate. You can calculate the dewpoint in insulation and use low perm insulation. Also can use a software like THERM. Insulation has to be thick enough for dew point to be within insulation.
there probably is a way to create a spreadsheet. You would have to manually enter the dewpooint. Or write a program in EES.

I would question your indoor conditions possible without extreme humidification or dehumidification. If outdoor is -40°C, I doubt a normal space will have 50% RH at 18°C unless it is a wet space. The same way in a climate with 30°C/RH 100% outdoor, it will be hard to have indoor RH at 50%.

It's correct about the hypothetical RH%. We should remove the indoor RH as a known too then:

1. Room in cold and dry climate

• Outdoor air: -40 C, 100% RH
• Indoor air: 18 C (electric heat and DX cooling)
• Duct: 500mm x 500mm, 5 m long, 200 L/s, -40 C supply air

2. Room in warm and wet climate

• Outdoor air: 30 C, 100% RH
• Indoor air: 18 C (electric heat and DX cooling)
• Duct: 500mm x 500mm, 5 m long, 200 L/s, 30 C supply air

Other than using software or writing a program, is there a more straightforward way to calculate it?

When the outdoor temperature is lower than the dew point of indoor temp, duct insulation is needed. Indoor relative humidity is needed to calculate the dew point, or you may estimate one, say 65%.

Your example 1 needs duct insulation, and example 2 does not.

I don't think it's a good idea to discharge outdoor air directly into the space. In most areas on earth, the outdoor conditions will change all year round, usually cold in winter. If you have ducted fan coils, connect the outside air to the return duct. If it is a ductless highwall unit, have the outside air discharge very close to the return grill.

I suggest you get more familiar with Heat Transfer and Psychometrics.

Seems to me that case 2 should be insulated as well, since the conditions stipulate could result in condensation on the inside of the duct, resulting in pooling of water at the bottom of the duct, which could lead to mold and other growth that could contaminate the makeup air.

TTFN (ta ta for now)
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