I may (or may not) be experiencing similar problems, starting yesterday. Very weird issues I can't resolve. Devices: N6P and Nexus 7 2013 wifi only; both on 6.0.1. Issues started on both devices simultaneously:
* Maildroid (MD) app will not contact or sync Gmail over wifi. No problem contacting/auto-syncing Yahoo or Roadrunner email over wifi. When I turn off wifi on the 6P, Gmail syncs in MD immediately over 4G. No 4G on the N7 so I tried the Gmail app - it will contact/sync when prompted, but will not auto-sync even though Settings > Accounts is set for auto-sync. Very, very weird - why only Gmail?
* No push notifications - Bleacher Report, NBA app will not pull push notifications over wifi. Turn wifi off and they immediately pop in over 4G. Apps themselves work fine over wifi. Push notifications are the only problem.
Notes: I haven't installed any new apps before this occurred. I cleared the system cache on the N7 but problem continues. I've confirmed the Gmail Imap settings are correct in MD (and, duh, they work on 4G). No problem syncing Gmail in Thunderbird on a Macbook Pro over same wifi.
Unlike the OP, I have no problem with Play Store, Youtube or any other app over wifi, so I'm not even sure my issue is the same. But it's similar and very puzzling. Not my intention to hijack this thread, but the similarity was such that I didn't want to start a separate one. Any suggestions/help would be appreciated.