There are two types of iPods.
The iPod touch runs iOS, and is similar to an iPhone that happens to lack GPS and the ability to connect to cellular voice and data service. Apple Music subscription songs might work on an iPod touch, assuming that its version of iOS is recent enough.
Other iPods run different operating systems, and they were on the market long before the Apple Music subscription service came along. While they can play the DRMed songs that the iTunes Store originally sold (until 2009), I would not expect them to be able to handle the DRM on Apple Music subscription songs, or to be able to play such songs. They wouldn't be able to check the Internet to see if a subscription had expired, and might not have any support for enforcing an expiration date based even on a local real-time clock.
I would suggest double-checking the songs that are causing grief to see what category they fall into:
- DRMed iTunes Store songs purchased in 2009 or earlier will have a "Protected" file type and will be encoded at a rate of 128 Kbps.
- iTunes Store songs purchased since then will be DRM-free and will be encoded at a rate of 256 Kbps.
- Apple Music subscription songs will have DRM, but will likely be encoded as 256 Kbps or as Apple Lossless (that is, encoded in higher quality than the original iTunes Store songs).