What is the maximum length of a Push Notification alert text?

It should be 236 bytes. There is no restriction on the size of the alert text as far as I know, but only the total payload size. So considering if the payload is minimal and only contains the alert information, it should look like:

{"aps":{"alert":""}}

That takes up 20 characters (20 bytes), leaving 236 bytes to put inside the alert string. With ASCII that will be 236 characters, and could be lesser with UTF8 and UTF16.

The real limits for the alert text are not documented anywhere. The only thing the documentation says is:

In iOS 8 and later, the maximum size allowed for a notification payload is 2 kilobytes; Apple Push Notification Service refuses any notification that exceeds this limit. (Prior to iOS 8 and in OS X, the maximum payload size is 256 bytes.)

This is what I could find doing some experiments.

• Alerts: Prior to iOS 7, the alerts display limit was 107 characters. Bigger messages were truncated and you would get a "..." at the end of the displayed message. With iOS 7 the limit seems to be increased to 235 characters. If you go over 8 lines your message will also get truncated.
• Banners: Banners get truncated around 62 characters or 2 lines.
• Notification Center: The messages in the notification center get truncated around 110 characters or 4 lines.
• Lock Screen: Same as a notification center.

Just as a reminder here is a very good note from the official documentation:

If necessary, iOS truncates your message so that it fits well in each notification delivery style; for best results, you shouldn’t truncate your message.