It looks like you are using some old version of the library (from 2012).
The latest version, which does not contain the problematic "extern" block, can be found here: https://github.com/adafruit/DHT-sensor-library
(The default example works nicely with my Arduino Nano v3.0 and DHT22 sensor)
Answer from Viliam Simko on Stack OverflowAvr/io.h: No such file or directory error when using Adafruit BNO08x library
c - Why this avr/io.h header give me fatal error? - Stack Overflow
Fatal error: avr/io.h: No such file or directory
avr/io.h not found
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I'm happy enough with that error as Tumbleweed installs the package in '/usr/avr/sys-root' so the header file is in /usr/avr/sys-root/include/avr/io.h
Rather then trying to change the Makefile, I struggle to read makefiles, I decided I'd simply put a link to where the include directory should be so:
ln -s /usr/avr/sys-root /usr/lib/avr
So now the header file is where you'd expect it to be '/usr/lib/avr/include/avr/io.h'
Trouble is that the qmk_firmware build still ain't happy with that. Perhaps it don't like links, but I thought that was transparent. Permissions on the directories and files are all good but still no jot.
If anybody could suggest a solution I'd be very grateful, failing that I'll have to install Ubuntu 16.04 somewhere.