What major tectonic events occur in Oligocene and Miocene time that might influence the climate?

Answer 1

The complete separation of Gondwana in the early Oligocene, so that Antarctica became surrounded by ocean and the Antarctic Circumpolar Current started, with global climatic consequences, eg. icing over Antarctica.

Also, the continued collision and uplift of the Himalayas. The Himalayas have a major climatic influence on central asia (drier) to south-east asia (warmer).

The uplift of the Andes in the late Miocene would have had climatic effects in South America - east Pacific.

Answer 2

The many major bentonite beds and other volcanic features from this age indicate that ash and volcanic gasses in the atmosphere likely had some impact on the climate.

Other tectonic activity uplifting areas would have had a more localized affect on climate, mainly precipitation. This would include continued activity in several mountain ranges.