OK, I was in contact with AirVPN support who were asking for the version of OpenVPN (it's 2.5.8 in DSM 7.2) and also for the OpenVPN log. I couldn't find it and, while googling the subject, came across a post by someone having similar problems with a different VPN provider.
This worked for me:
- Go to the config generator
- Select Router
- Don't click on Advanced, just select UDP 443 (I believe that's selected by default)
- Select your server/country/whatever you like
- Generate the ovpn file
In DSM:
- Create a VPN profile
- Give it a name
- Enter your AirVPN credentials
- Select the ovpn file you just generated
- Click the checkboxes in the next screen
- Click Connect and it should connect... worked here anyway.
This had been driving me nuts for a few hours. I even tried another VPN (free trial) which didn't work, so I was really wondering if the server's VPN module was broken!