I haven't done the calculations on this, but I think viewing this from a standpoint of energies would get you a long way.
Your example with the tea and the milk is true because the milk absorbs some of the energy of the hot tea. Essentially:
Tea is hot relative to surroundings and milk
Larger difference in temperature = larger flow of heat = faster cooling down
Adding milk at the start absorbs energy (temperature) from hot tea, slows the cooling process
Adding milk at the end absorbs less energy, but cooling process has run at maximum speed until then
Net result is faster cooling down when milk is added at the end
The blowing fan with the wet laundry works a bit differently.
Laundry is humid compared to surroundings
Larger difference in humidity = higher rate of evaporation = faster drying
Turning on the fan at the start blows away humid air near the laundry, increasing the humid-dry gradient near the laundry
Drying process speeds up
Turning on the fan at the end blows away humid air near the laundry (but less than when it was more wet), increasing the humid-dry gradient but less so than when it was more wet
Drying process speeds up less
Net result is faster drying when fan is turned on at the start
I am assuming the fan blows air at room temperature. If it blows hot air the result is the same, but you won't get that cuddly soft and warm feeling laundry at the end. So if you want that, turn on the hot air near the end of the drying process and cozy up to your warm dry laundry with a cup of hot chocolate.