bperrybap:
That assumes you have the UTC and local time.
which is exactly what the APIs provide.
bperrybap:
If that is the case great, but they must have some sort of crazy database that can map GPS coordinates (or cities) to get the appropriate DST rules to take that into account.
You enter the city or postal code, or a spot on the map and you get both. Daylight Savings and time zones are Political, not geographical, after all.
After all, what would be the point of using a weather API without getting to select your locality?
bperrybap:
And around the World things can be quite different.
yes, they could be... but they really aren't. So, if they are changing, they are events that people know about well in advance. Let's face it, governments don't make emergency changes to their timezones! 
bperrybap:
And that is the point I'm making. If the API can provide an accurate local time, it must be able to take into consideration all the messy details of daylight savings time in various locations.
If they can handle the "messy details" about the atmosphere's pressure, temperature, ozone, humidity, pollution, windspeed, etc. outside your home, I think DST would be rather boring for these guys. 
Using the APIs correctly make your concerns a tempest in a teacup.