Irish weather is normally cool and wet in October.
The Irish weather is normally cool and wet in October.
are perfectly normal and grammatical. Without the article, "Irish weather" means "all weather in Ireland". There might be multiple weather patterns, but the speaker is saying that they share this characteristic. In this case the meaning is unchanged.
In general at the start of a sentence "The weather" is more common, but can be replaced by an adjective indicating what whether is meant, such as "Irish weather"
The weather is expected to remain clear for the next few days.
When speaking of weather in general or in a particular area (often implied by context) an article is usual in this construction. However
Clear weather is expected to persist for the next few days.
would not use an article, and is perfectly normal. Tis is much like your example:
Fair weather was forecast for the following day.
This could be recast as
The weather was forecast to be fair for the following day.
About the rescue example in the question:
- The rescue operation has been complicated by bad weather.
- The rescue operation has been complicated by the bad weather.
The meaning is not much different between these two. The first might refer to multiple instances of bad weather, or bad weather in several different places. The second emphasized "the bad weather" as a unitary thing to a greater degree. Both are grammatical and common.